Sharp Functional Inequalities for Bounded Turning Functions Associated with a Single-Lobed Elliptic Domain

1. Introduction

Let $\mathsf{\Theta }$ represent the family of analytic functions $\kappa$ defined in the open unit disk $\mathsf{\Delta }=\{\xi \in \mathbb{C}:|\xi |<1\}$ that satisfying $\kappa \left(0\right)={\kappa }^{\prime }\left(0\right)-1=0$ and admit the following power series expansion:

$$\begin{array}{c}\hfill \kappa \left(\xi \right)=\xi +{\displaystyle \sum _{j=2}^{\infty }}{\alpha }_j{\xi }^j.\end{array}$$

(1)

Let $\mathcal{S}$ denote the class of all functions of $\mathsf{\Theta }$ that are univalent in $\mathsf{\Delta }.$ Also, let

$$\begin{array}{c}\hfill \mathcal{R}=\left\{\kappa \in \mathsf{\Theta }:Re{\kappa }^{\prime }\left(\xi \right)>0,\xi \in \mathsf{\Delta }\right\},\end{array}$$

$$\begin{array}{c}\hfill {\mathcal{S}}^{*}=\left\{\kappa \in \mathsf{\Theta }:Re{\displaystyle \frac{\xi {\kappa }^{\prime }\left(\xi \right)}{\kappa \left(\xi \right)}}>0,\xi \in \mathsf{\Delta }\right\},\end{array}$$

denote the classes of bounded turning and starlike functions, respectively.

For $p,j\in \mathbb{N},$ the Hankel determinant ${\mathcal{H}}_{p,j}\left(\kappa \right)$ of $\kappa \in \mathsf{\Theta }$ of the form (1) is defined as

$${\mathcal{H}}_{p,j}\left(\kappa \right)=\left|\begin{array}{cccc}{\alpha }_j& {\alpha }_{j+1}& ···& {\alpha }_{j+p-1}\\ {\alpha }_{j+1}& {\alpha }_{j+2}& ···& {\alpha }_{j+p}\\ ⋮& ⋮& ⋮& ⋮& \\ {\alpha }_{j+p-1}& {\alpha }_{j+p}& ···& {\alpha }_{j+2p-2}\end{array}\right|.$$

The Hankel determinant

$$\begin{array}{c}\hfill {\mathcal{H}}_{2,1}\left(\kappa \right)={\alpha }_3-{\alpha }_2^2\end{array}$$

(2)

is the recognized as the Fekete-Szegö functional [1]. The second Hankel determinant ${\mathcal{H}}_{2,2}\left(\kappa \right)$ is represented by

$$\begin{array}{c}\hfill {\mathcal{H}}_{2,2}\left(\kappa \right)={\alpha }_2{\alpha }_4-{\alpha }_3^2.\end{array}$$

(3)

The third Hankel determinant ${\mathcal{H}}_{3,1}\left(\kappa \right)$ is represented by

$$\begin{array}{c}\hfill {\mathcal{H}}_{3,1}\left(\kappa \right)=-{\alpha }_3^3+{\alpha }_5{\alpha }_3-{\alpha }_5{\alpha }_2^2+2{\alpha }_4{\alpha }_3{\alpha }_2-{\alpha }_4^2.\end{array}$$

(4)

Babalola [2] was the first person to study the upper bound of ${\mathcal{H}}_{3,1}\left(\kappa \right)$ for subclass of $\mathcal{S}.$ For more details on this topic readers are advised to see the work of several researchers [3,4,5,6,7,8,9,10,11,12].

Let $\kappa \in \mathcal{S}$. The logarithmic coefficients, introduced through the expansion

$$log{\displaystyle \frac{\kappa \left(\xi \right)}{\xi }}=2{\displaystyle \sum _{j=1}^{\infty }}{\vartheta }_j{\xi }^j,(\xi \in \mathsf{\Delta })$$

(5)

have attracted considerable attention for their role in characterizing the geometric properties of univalent functions. ${\vartheta }_j$ are called logarithmic coefficients [13]. By differentiating (5) and the equating coefficients we obtain

$$\left\{\begin{array}{c}{\vartheta }_1={\displaystyle \frac{1}{2}}{\alpha }_2,\hfill \\ {\vartheta }_2={\displaystyle \frac{1}{2}}({\alpha }_3-{\displaystyle \frac{1}{2}}{\alpha }_2^2),\hfill \\ {\vartheta }_3={\displaystyle \frac{1}{2}}({\alpha }_4-{\alpha }_2{\alpha }_3+{\displaystyle \frac{1}{3}}{\alpha }_2^3),\hfill \\ {\vartheta }_4={\displaystyle \frac{1}{2}}({\alpha }_5-{\displaystyle \frac{1}{2}}{\alpha }_3^2+{\alpha }_3{\alpha }_2^2-{\alpha }_4{\alpha }_2-{\displaystyle \frac{1}{4}}{\alpha }_2^4).\hfill \\ \hfill \end{array}\right.$$

(6)

For $\kappa \in \mathcal{S}$, the inverse function ${\kappa }^{-1}$ is given by

$${\kappa }^{-1}\left(w\right)=w+{\displaystyle \sum _{j=2}^{\infty }}{A}_j{w}^j\left(\right|w|<{\displaystyle \frac{1}{4}}).$$

(7)

From (7), we have

$$\left\{\begin{array}{c}{A}_2=-{\alpha }_2,\hfill \\ A_3=2{\alpha }_2^2-{\alpha }_3,\hfill \\ A_4=-5{\alpha }_2^3+5{\alpha }_2{\alpha }_3-{\alpha }_4,\hfill \\ A_5=14{\alpha }_2^4-21{\alpha }_2^2{\alpha }_3+6{\alpha }_2{\alpha }_4+3{\alpha }_3^2-{\alpha }_5.\hfill \\ \hfill \end{array}\right.$$

(8)

For ${\kappa }^{-1}\in \mathcal{S},$ the logarithmic inverse coefficients ${\delta }_j,j\in \mathbb{N}$ are defined by the equation

$$log{\displaystyle \frac{{\kappa }^{-1}\left(w\right)}{w}}=2{\displaystyle \sum _{j=1}^{\infty }}{\delta }_j{w}^j\left(\right|w|<{\displaystyle \frac{1}{4}}).$$

(9)

By using (8) and differentiating the equation (9), we have

$$\left\{\begin{array}{c}{\delta }_1=-{\displaystyle \frac{1}{2}}{\alpha }_2,\hfill \\ {\delta }_2=-{\displaystyle \frac{1}{4}}(2{\alpha }_3-3{\alpha }_2^2),\hfill \\ {\delta }_3=-{\displaystyle \frac{1}{6}}(3{\alpha }_4-12{\alpha }_2{\alpha }_3+10{\alpha }_2^3),\hfill \\ {\delta }_4=-{\displaystyle \frac{1}{8}}(4{\alpha }_5-20{\alpha }_4{\alpha }_2+60{\alpha }_3{\alpha }_2^2-10{\alpha }_3^2-35{\alpha }_2^4).\hfill \\ \hfill \end{array}\right.$$

(10)

The Schwarzian derivative for $\kappa \in \mathsf{\Theta }$ is defined by

$$\begin{array}{c}\hfill S\left(\kappa \right)\left(\xi \right)={\left({\displaystyle \frac{{\kappa }^{\prime \prime }\left(\xi \right)}{{\kappa }^{\prime }\left(\xi \right)}}\right)}^{\prime }-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{{\kappa }^{\prime \prime }\left(\xi \right)}{{\kappa }^{\prime }\left(\xi \right)}}\right)}^2(\xi \in \mathsf{\Delta }).\end{array}$$

The higher-order Schwarzian derivative ${\eta }_m\left(\kappa \right)$ are defined inductively (see [14,15]) as follows:

$$\begin{array}{c}\hfill {\eta }_3\left(\kappa \right)=S\left(k\right),\end{array}$$

(11)

$$\begin{array}{c}\hfill {\eta }_{m+1}\left(\kappa \right)={\left({\eta }_m\left(\kappa \right)\right)}^{\prime }-(m-1){\eta }_m\left(\kappa \right){\displaystyle \frac{{\kappa }^{\prime \prime }}{{\kappa }^{\prime }}},m\ge 3.\end{array}$$

(12)

Let $S_m={\eta }_m\left(\kappa \right)\left(0\right).$ If $\kappa \in \mathsf{\Theta }$ is of the form (1), then

$$\left\{\begin{array}{c}{S}_3=6({\alpha }_3-{\alpha }_2^2),\hfill \\ S_4=24({\alpha }_4-3{\alpha }_2{\alpha }_3+2{\alpha }_2^3),\hfill \\ S_5=24(5{\alpha }_5-20{\alpha }_2{\alpha }_4-9{\alpha }_3^2+48{\alpha }_2^2{\alpha }_3-24{\alpha }_2^4).\hfill \\ \hfill \end{array}\right.$$

(13)

Let $\kappa \in \mathcal{S}$ and ${\sigma }_m={\eta }_m\left({\kappa }^{-1}\right)\left(0\right).$ From (8) and (13), we have

$$\left\{\begin{array}{c}{\sigma }_3=6(-{\alpha }_3+{\alpha }_2^2),\hfill \\ {\sigma }_4=24(-{\alpha }_4+2{\alpha }_2{\alpha }_3-{\alpha }_2^3),\hfill \\ {\sigma }_5=24(-5{\alpha }_5+6{\alpha }_3^2+10{\alpha }_2{\alpha }_4-17{\alpha }_2^2{\alpha }_3+6{\alpha }_2^4).\hfill \\ \hfill \end{array}\right.$$

(14)

In 2025, Tayyah et al. [16] introduced and studied a subclass of starlike functions connected to a single-lobed elliptic domain.

Definition 1 ([16])If a function $\kappa \in \mathsf{\Theta }$ has the form given in (1), then it belongs to the subclass ${\mathcal{S}}_{LE}^{*}$, provided the following conditions are satisfied:

$$\begin{array}{c}\hfill {\displaystyle \frac{\xi {\kappa }^{\prime }\left(\xi \right)}{\kappa \left(\xi \right)}}\prec {\Phi }_{LE},(\xi \in \mathsf{\Delta }),\end{array}$$

where the function

$$\begin{array}{c}\hfill {\Phi }_{LE}=2-\sqrt{{\displaystyle \frac{2-{\xi }^2}{2(1+si{n}^{-1}\left(\frac{2\xi }{3}\right))}}}=1+{\displaystyle \frac{\xi }{3}}+{\displaystyle \frac{{\xi }^2}{12}}+{\displaystyle \frac{11{\xi }^3}{324}}-{\displaystyle \frac{5{\xi }^4}{864}}+···\end{array}$$

(15)

is univalent.

Definition 2Let $\kappa \in \mathsf{\Theta }$, given in (1). Then $\kappa \in {\mathcal{R}}_{LE}$ if it satisfies the condition

$$\begin{array}{c}\hfill {\kappa }^{\prime }\left(\xi \right)\prec {\Phi }_{LE},(\xi \in \mathsf{\Delta }).\end{array}$$

For the class ${\mathcal{S}}_{LE}^{*}$, Tayyah [16] obtained the following theorem.

Theorem A ([16]) (1) If $\kappa \in {\mathcal{S}}_{LE}^{*}$, then $|{\alpha }_2|\le {\displaystyle \frac{1}{3}},|{\alpha }_3|\le {\displaystyle \frac{1}{6}},|{\alpha }_4|\le {\displaystyle \frac{1}{9}},\left|{\alpha }_5\right|\le {\displaystyle \frac{1}{12}};$

(2) If $\kappa \in {\mathcal{S}}_{LE}^{*}$, then $|{\mathcal{H}}_{2,2}|\le {\displaystyle \frac{1}{36}}$, $|{\mathcal{H}}_{3,1}|\le {\displaystyle \frac{1}{81}}$.

In this paper, we initially established some bounds on the coefficients of a subclass of bounded turning functions, denoted as ${\mathcal{R}}_{LE}$, which are associated with a single-lobed elliptic domain. For functions belonging to this class, we determined the precise upper bounds for the second and third Hankel determinants, as well as the upper bound for the Hankel determinant with logarithmic coefficients. Additionally, we examined the upper bounds for higher-order Schwarzian derivatives. Furthermore, we also investigated the upper bounds for the corresponding coefficients of the inverse functions.

Of particular interest is the Schwarz functions class $\mathcal{B}$, consisting of functions of the form

$$\beta \left(\xi \right)={\beta }_1\xi +{\beta }_2{\xi }^2+{\beta }_3{\xi }^3+{\beta }_4{\xi }^4+···,(\xi \in \mathsf{\Delta }),$$

(16)

which satisfy $\left|\beta \right(\xi \left)\right|<1$ in $\mathsf{\Delta }$ and $\beta \left(0\right)=0.$

The following set of Lemmas are vital to our main results.

Lemma 1.1 ([17]) If $\kappa \in \mathcal{B}$, then $|{\beta }_j|\le 1(j\ge 1)$.

Lemma 1.2 ([18]) Let $\kappa \in \mathcal{B}$ is given by (16) and $\gamma \in \mathbb{C}$. Then

$$\begin{array}{c}\hfill |{\beta }_2+\gamma {\beta }_1^2|\le max\{1,\left|\gamma \right|\}.\end{array}$$

Lemma 1.3 ([19]) If $\kappa \in \mathcal{B}$, then for any real numbers ν and ρ such that

$$\begin{array}{c}\hfill (\nu ,\rho )\in \{(\nu ,\rho ):|\nu |\le {\displaystyle \frac{1}{2}},-1\le \rho \le 1\}\cup \{(\nu ,\rho ):{\displaystyle \frac{1}{2}}\le \left|\nu \right|\le 2,{\displaystyle \frac{4}{27}}{\left(\right|\nu |+1)}^3-\left(\right|\nu |+1)\le \rho \le 1\},\end{array}$$

the following sharp estimates holds

$$\begin{array}{c}\hfill |{\beta }_3+\nu {\beta }_1{\beta }_2+\rho {\beta }_1^3|\le 1.\end{array}$$

Lemma 1.4 ([20]) Let $\kappa \in \mathcal{B}$, $\varkappa \in \mathbb{C}$ and $\left|\varkappa \right|\le 1.$ Then

$$\begin{array}{c}\hfill |{\beta }_4+2\varkappa {\beta }_1{\beta }_3+3{\varkappa }^2{\beta }_1^2{\beta }_2+\varkappa {\beta }_2^2+{\varkappa }^3{\beta }_1^4|\le 1.\end{array}$$

Lemma 1.5 ([21]) If $\kappa \in \mathcal{B}$, then

$$|{\beta }_2|\le 1-|{\beta }_1{|}^2,|{\beta }_4|\le 1-|{\beta }_1{|}^2-{\left|{\beta }_2\right|}^2.$$

Lemma 1.6 ([22]) Let $\kappa \in \mathcal{B}$, then

$$\begin{array}{ccc}\hfill {\beta }_1& =& {\zeta }_1,\hfill \\ \hfill {\beta }_2& =& (1-|{\zeta }_1{|}^2){\zeta }_2,\hfill \\ \hfill {\beta }_3& =& -(1-|{\zeta }_1{|}^2)\overline{{\zeta }_1}{\zeta }_2^2+(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2){\zeta }_3\hfill \\ \hfill {\beta }_4& =& (1-|{\zeta }_1{|}^2){\overline{{\zeta }_1}}^2{\zeta }_2^3-(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2)(\overline{{\zeta }_2}{\zeta }_3^2+2\overline{{\zeta }_1}{\zeta }_2{\zeta }_3)\hfill \\ & +& (1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_3{|}^2){\zeta }_4.\hfill \end{array}$$

for some ${\zeta }_m$ with $|{\zeta }_m|\le 1(m=1,2,3,4).$

2. The Sharp Functional Inequalities for the Class ${\mathcal{R}}_{LE}$

In the field of Geometric function theory, one of the classical conjecture proposed by Lawrence Zalcman in 1960 is that the coefficients of class $\mathcal{S}$ satisfy the inequality,

$$\begin{array}{c}\hfill |{\alpha }_j^2-{\alpha }_{2j-1}{|\le (j-1)}^2.\end{array}$$

The above form holds equality only for the famous Koebe function $k={\displaystyle \frac{\xi }{{(1-\xi )}^2}}$ and its rotation. In 1999, Ma [23] proposed a generalized Zalcman conjecture for $\kappa \in \mathcal{S}$ that

$$\begin{array}{c}\hfill |{\alpha }_i{\alpha }_j-{\alpha }_{i+j-1}|\le (i-1)(j-1)\end{array}$$

which is still an open problem.

Firstly, we evaluate the second and third-order Hankel determinants. Further we evaluate the generalized Zalcman functional for this class.

Theorem 2.1If the function $\kappa \in {\mathcal{R}}_{LE}$ and be given by equation (1). Then

$$|{\alpha }_2|\le {\displaystyle \frac{1}{6}},|{\alpha }_3|\le {\displaystyle \frac{1}{9}},|{\alpha }_4|\le {\displaystyle \frac{1}{12}},\left|{\alpha }_5\right|\le {\displaystyle \frac{1}{15}}.$$

These bounds are sharp for the functions defined below, respectively

$$\begin{array}{c}\hfill {\kappa }_1\left(\xi \right)={\int }_0^{\xi }\left(2-\sqrt{{\displaystyle \frac{2-t^2}{2(1+si{n}^{-1}\left(2t\right)/3)}}}\right)dt=\xi +{\displaystyle \frac{1}{6}}{\xi }^2+···,\end{array}$$

(17)

$$\begin{array}{c}\hfill {\kappa }_2\left(\xi \right)={\int }_0^{\xi }\left(2-\sqrt{{\displaystyle \frac{2-t^4}{2(1+si{n}^{-1}\left(2t^2\right)/3)}}}\right)dt=\xi +{\displaystyle \frac{1}{9}}{\xi }^3+···,\end{array}$$

(18)

$$\begin{array}{c}\hfill {\kappa }_3\left(\xi \right)={\int }_0^{\xi }\left(2-\sqrt{{\displaystyle \frac{2-t^6}{2(1+si{n}^{-1}\left(2t^3\right)/3)}}}\right)dt=\xi +{\displaystyle \frac{1}{12}}{\xi }^4+···,\end{array}$$

(19)

$$\begin{array}{c}\hfill {\kappa }_4\left(\xi \right)={\int }_0^{\xi }\left(2-\sqrt{{\displaystyle \frac{2-t^8}{2(1+si{n}^{-1}\left(2t^4\right)/3)}}}\right)dt=\xi +{\displaystyle \frac{1}{15}}{\xi }^5+···.\end{array}$$

(20)

Proof Supposing that $\kappa \in {\mathcal{R}}_{LE}$ has the form (1), then there exists a function $\beta$ analytic in $\mathsf{\Delta }$ with $\left|\beta \right(\xi \left)\right|<1$ and $\beta \left(0\right)=0,$ satisfying

$$\begin{array}{c}\hfill {\kappa }^{\prime }\left(\xi \right)=2-\sqrt{{\displaystyle \frac{2-{\beta }^2\left(\xi \right)}{2(1+si{n}^{-1}\left(\frac{2\beta \left(\xi \right)}{3}\right))}}}.\end{array}$$

(21)

From (1), we have

$$\begin{array}{cc}\hfill {\kappa }^{\prime }\left(\xi \right)& =1+2{\alpha }_2\xi +3{\alpha }_3{\xi }^2+4{\alpha }_4{\xi }^3+5{\alpha }_5{\xi }^4+···.\hfill \end{array}$$

(22)

From (16) and (15), we obtain

$$\begin{array}{ccc}\hfill 2-\sqrt{{\displaystyle \frac{2-{\beta }^2\left(\xi \right)}{2(1+si{n}^{-1}\left(\frac{2\beta \left(\xi \right)}{3}\right))}}}& =\hfill & \hfill 1+{\displaystyle \frac{1}{3}}{\beta }_1\xi +({\displaystyle \frac{1}{3}}{\beta }_2+{\displaystyle \frac{1}{12}}{\beta }_1^2){\xi }^2+({\displaystyle \frac{1}{3}}{\beta }_3+{\displaystyle \frac{1}{6}}{\beta }_1{\beta }_2+{\displaystyle \frac{11}{324}}{\beta }_1^3){\xi }^3\\ \hfill & +\hfill & \hfill ({\displaystyle \frac{1}{3}}{\beta }_4+{\displaystyle \frac{1}{6}}{\beta }_1{\beta }_3+{\displaystyle \frac{1}{12}}{\beta }_2^2+{\displaystyle \frac{11}{108}}{\beta }_1^2{\beta }_2-{\displaystyle \frac{5}{864}}{\beta }_1^4){\xi }^4+···.\end{array}$$

(23)

Equating the first four coefficients of (22) and (23), we have

$$\begin{array}{c}\hfill {\alpha }_2={\displaystyle \frac{1}{6}}{\beta }_1,\end{array}$$

(24)

$$\begin{array}{c}\hfill {\alpha }_3={\displaystyle \frac{1}{9}}{\beta }_2+{\displaystyle \frac{1}{36}}{\beta }_1^2,\end{array}$$

(25)

$$\begin{array}{c}\hfill {\alpha }_4={\displaystyle \frac{1}{12}}{\beta }_3+{\displaystyle \frac{1}{24}}{\beta }_1{\beta }_2+{\displaystyle \frac{11}{1296}}{\beta }_1^3,\end{array}$$

(26)

$$\begin{array}{c}\hfill {\alpha }_5={\displaystyle \frac{1}{4320}}(288{\beta }_4+144{\beta }_1{\beta }_3+88{\beta }_1^2{\beta }_2+72{\beta }_2^2-5{\beta }_1^4).\end{array}$$

(27)

From (24), (25) Lemma 1.1 and 1.2, we have

$$|{\alpha }_2|\le {\displaystyle \frac{1}{6}},\left|{\alpha }_3\right|\le {\displaystyle \frac{1}{9}}.$$

Rearranging (26), we get

$$\begin{array}{c}\hfill |{\alpha }_4|={\displaystyle \frac{1}{12}}|{\beta }_3+{\displaystyle \frac{1}{2}}{\beta }_1{\beta }_2+{\displaystyle \frac{11}{108}}{\beta }_1^3|.\end{array}$$

Using Lemma 1.3 for $\nu ={\displaystyle \frac{1}{2}}$ and $\rho ={\displaystyle \frac{11}{108}}$, we have $|{\alpha }_4|\le {\displaystyle \frac{1}{12}}$.

From (27), Lemma 1.4 and 1.5, we have

$$\begin{array}{cc}\hfill |{\alpha }_5|& ={\displaystyle \frac{1}{4320}}|72({\beta }_4+2{\beta }_3{\beta }_1+3{\beta }_2{\beta }_1^2+{\beta }_2^2+{\beta }_1^4)+216{\beta }_4-128{\beta }_2{\beta }_1^2-77{\beta }_1^4|\hfill \\ \hfill & \le {\displaystyle \frac{1}{4320}}[72+216(1-|{\beta }_1{|}^2-|{\beta }_2{|}^2)+128|{\beta }_1{|}^2(1-|{\beta }_1{|}^2)+77|{\beta }_1{|}^4]\hfill \\ \hfill & ={\displaystyle \frac{1}{4320}}[288-88|{\beta }_1{|}^2-216|{\beta }_2{|}^2-51|{\beta }_1{|}^4]\le {\displaystyle \frac{288}{4320}}={\displaystyle \frac{1}{15}}.\hfill \end{array}$$

Conjecture 2.1Let $\kappa \in {\mathcal{R}}_{LE}$. Then

$$|{\alpha }_j|\le {\displaystyle \frac{1}{3j}},j=2,3,···.$$

Using equations (24), (25) and Lemma 1.2, we easily get the following theorem.

Theorem 2.2If a function κ of the form equation (1) belongs to ${\mathcal{R}}_{LE}$ then

$$|{\alpha }_3-\gamma {\alpha }_2^2|\le {\displaystyle \frac{1}{9}}max\{1,{\displaystyle \frac{|1-\gamma |}{4}}\}.$$

This result is sharp.

Theorem 2.3Suppose $\kappa \in {\mathcal{R}}_{LE}$ is be given (1). Then

$$|{\alpha }_2{\alpha }_4-{\alpha }_3^2|\le {\displaystyle \frac{1}{81}}.$$

The equality is sharp for the function ${\kappa }_2\left(\xi \right).$

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (24), (25), (26) and (3), we achieve

$$\begin{array}{c}\hfill {\mathcal{H}}_{2,2}\left(\kappa \right)={\displaystyle \frac{1}{7776}}(108{\beta }_1{\beta }_3+6{\beta }_1^2{\beta }_2+5{\beta }_1^4-96{\beta }_2^2).\end{array}$$

(28)

From Lemma 1.6 and (28), we yield

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{2,2}\left(\kappa \right)& =& {\displaystyle \frac{1}{7776}}[-108|{\zeta }_1{|}^2(1-|{\zeta }_1{|}^2){\zeta }_2^2+108{\zeta }_1(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2){\zeta }_3+6{\zeta }_1^2(1-|{\zeta }_1{|}^2){\zeta }_2+5{\zeta }_1^4\hfill \\ & -& 96(1-|{\zeta }_1{|}^2{)}^2{\zeta }_2^2].\hfill \end{array}$$

By setting $|{\zeta }_1|=\zeta ,|{\zeta }_2|=ϵ$ and upon taking $|{\zeta }_3|\le 1$, we obtain

$$\begin{array}{ccc}\hfill |{\mathcal{H}}_{2,2}\left(\kappa \right)|& \le & {\displaystyle \frac{1}{7776}}[108{\zeta }^2(1-{\zeta }^2){ϵ}^2+108\zeta (1-{\zeta }^2)(1-{ϵ}^2)+6{\zeta }^2(1-{\zeta }^2)ϵ+5{\zeta }^4+96{(1-{\zeta }^2)}^2{ϵ}^2]\hfill \\ & =& {\displaystyle \frac{1}{7776}}{{\rm Y}}_1(\zeta ,ϵ).\hfill \end{array}$$

Differentiating ${{\rm Y}}_1(\zeta ,ϵ)$ with respect to $ϵ$, we yield

$${\displaystyle \frac{\partial {{\rm Y}}_1}{\partial ϵ}}=6(1-{\zeta }^2){\zeta }^2+24(1-{\zeta }^2)({\zeta }^2-9\zeta +8)ϵ\ge 0.$$

Obviously, ${{\rm Y}}_1$ attains its maximum value at $ϵ=1$. So, we get

$$\begin{array}{ccc}\hfill |{\mathcal{H}}_{2,2}\left(\kappa \right)|& \le & {\displaystyle \frac{1}{7776}}{{\rm Y}}_1(\zeta ,1)={\displaystyle \frac{1}{7776}}(96-78{\zeta }^2-13{\zeta }^4)\le {\displaystyle \frac{96}{7776}}={\displaystyle \frac{1}{81}}.\hfill \end{array}$$

Theorem 2.4If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\mathcal{H}}_{3,1}\left(\kappa \right)|\le {\displaystyle \frac{1}{144}}.$$

The equality holds for the function ${\kappa }_3\left(\xi \right)$

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (4), (24), (25), (26) and (27), we achieve

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{3,1}\left(\kappa \right)& =& {\displaystyle \frac{1}{8398080}}(4032{\beta }_2^3+8748{\beta }_1^2{\beta }_2^2-3300{\beta }_1^4{\beta }_2-125{\beta }_1^6+62208{\beta }_2{\beta }_4-1296{\beta }_1{\beta }_2{\beta }_3\hfill \\ & -& 5400{\beta }_1^3{\beta }_3-58320{\beta }_3^2).\hfill \end{array}$$

(29)

From (29) and Lemma 1.6, we achieve

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{3,1}\left(\kappa \right)& =& {\displaystyle \frac{1}{8398080}}[-125{\zeta }_1^6+8748(1-|{\zeta }_1{|}^2{)}^2{\zeta }_1^2{\zeta }_2^2-3300(1-|{\zeta }_1{|}^2){\zeta }_1^4{\zeta }_2+62208(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_3{|}^2)\hfill \\ & \times & (1-|{\zeta }_2{|}^2){\zeta }_2{\zeta }_4+4032(1-|{\zeta }_1{|}^2{)}^3{\zeta }_2^3-1296{\zeta }_1{\zeta }_2(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2){\zeta }_3+1296(1-|{\zeta }_1{|}^2{)}^2{\left|{\zeta }_1\right|}^2{\zeta }_2^3\hfill \\ & -& 5400(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2){\zeta }_1^3{\zeta }_3+5400(1-|{\zeta }_1{|}^2){\zeta }_1^3\overline{{\zeta }_1}{\zeta }_2^2-62208|{\zeta }_2{|}^2(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{{|}^2)}^2{\zeta }_3^2\hfill \\ & -& 58320(1-|{\zeta }_2{|}^2{)}^2(1-|{\zeta }_1{|}^2{)}^2{\zeta }_3^2-7776\overline{{\zeta }_1}{\zeta }_2^2(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2){\zeta }_3+3888{\overline{{\zeta }_1}}^2{\zeta }_2^4(1-|{\zeta }_1{{|}^2)}^2].\hfill \end{array}$$

Thus, we achieve

$${\mathcal{H}}_{3,1}\left(\kappa \right)={\displaystyle \frac{1}{8398080}}\left[{\tau }_1\left({\zeta }_1,{\zeta }_2\right)+{\tau }_2\left({\zeta }_1,{\zeta }_2\right){\zeta }_3+{\tau }_3\left({\zeta }_1,{\zeta }_2\right){\zeta }_3^2+{ϝ}_1\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right){\zeta }_4\right],$$

where

$$\begin{array}{ccc}\hfill {\tau }_1\left({\zeta }_1,{\zeta }_2\right)& =& \left(1-|{\zeta }_1{|}^2\right)\left\{[8748{\zeta }_1^2{\zeta }_2^2+(4032-2736|{\zeta }_1{|}^2){\zeta }_2^3+3888{\overline{{\zeta }_1}}^2{\zeta }_2^4\left]\right(1-|{\zeta }_1{|}^2)+5400{\zeta }_2^2{\zeta }_1^3\overline{{\zeta }_1}\right.\hfill \\ & & \left.-3300{\zeta }_2{\zeta }_1^4\right\}-125{\zeta }_1^6,\hfill \\ \hfill {\tau }_2\left({\zeta }_1,{\zeta }_2\right)& =& (1-|{\zeta }_2{|}^2)\left(1-|{\zeta }_1{|}^2\right)\left[(-1296{\zeta }_1{\zeta }_2-7776\overline{{\zeta }_1}{\zeta }_2^2)(1-|{\zeta }_1{|}^2)-5400{\zeta }_1^3\right],\hfill \\ \hfill {\tau }_3\left({\zeta }_1,{\zeta }_2\right)& =& (1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2\left)\right(-58320-3888|{\zeta }_2{|}^2),\hfill \\ \hfill {ϝ}_1\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)& =& 62208(1-|{\zeta }_3{|}^2\left)\right(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{{|}^2)}^2{\zeta }_2.\hfill \end{array}$$

Upon taking $|{\zeta }_4|\le 1$, and setting $\zeta =|{\zeta }_1|,ϵ=|{\zeta }_2|,\aleph =|{\zeta }_3|$, we yield

$$\begin{array}{ccc}\hfill \left|{\mathcal{H}}_{3,1}\left(\kappa \right)\right|& \le & {\displaystyle \frac{1}{8398080}}\left(\left|{\tau }_1\left({\zeta }_1,{\zeta }_2\right)\right|+\left|{\tau }_2\left({\zeta }_1,{\zeta }_2\right)\right|\varkappa +\left|{\tau }_3\left({\zeta }_1,{\zeta }_2\right)\right|{\aleph }^2+\left|{ϝ}_1\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)\right|\right)\hfill \\ & \le & {\displaystyle \frac{1}{8398080}}{\Psi }_1\left(\zeta ,ϵ,\aleph \right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\Psi }_1\left(\zeta ,ϵ,\aleph \right)={\varsigma }_1\left(\zeta ,ϵ\right)+{\varsigma }_2\left(\zeta ,ϵ\right)\aleph +{\varsigma }_3\left(\zeta ,ϵ\right){\aleph }^2+{\varsigma }_4\left(\zeta ,ϵ\right)(1-{\aleph }^2),\end{array}$$

with

$$\begin{array}{ccc}\hfill {\varsigma }_1\left(\zeta ,ϵ\right)& =& \left(1-{\zeta }^2\right)\left\{\left[(3888{\zeta }^2{ϵ}^4+(4032-2736{\zeta }^2){ϵ}^3+8748{\zeta }^2{ϵ}^2\right](1-{\zeta }^2)+5400{\zeta }^4{ϵ}^2+3300{\zeta }^4ϵ\right\}+125{\zeta }^6,\hfill \\ \hfill {\varsigma }_2\left(\zeta ,ϵ\right)& =& \left(1-{ϵ}^2\right)(1-{\zeta }^2)[(1296\zeta ϵ+7776\zeta {ϵ}^2)(1-{\zeta }^2)+5400{\zeta }^3],\hfill \\ \hfill {\varsigma }_3\left(\zeta ,ϵ\right)& =& \left(1-{ϵ}^2\right){(1-{\zeta }^2)}^2(3888{ϵ}^2+58320),\hfill \\ \hfill {\varsigma }_4\left(\zeta ,ϵ\right)& =& 62208{(1-{\zeta }^2)}^2(1-{ϵ}^2)ϵ.\hfill \end{array}$$

Differentiating ${\Psi }_1(\zeta ,ϵ,\aleph )$ with respect to ℵ, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\Psi }_1}{\partial \aleph }}& =& {\varsigma }_2(\zeta ,ϵ)+2[{\varsigma }_3(\zeta ,ϵ)-{\varsigma }_4(\zeta ,ϵ)]\aleph \hfill \\ & & ={\varsigma }_2(\zeta ,ϵ)+7776(1-{ϵ}^2)({ϵ}^2-16ϵ+15){(1-{\zeta }^2)}^2\aleph \ge 0.\hfill \end{array}$$

Therefore, ${\Psi }_1(\zeta ,ϵ,\aleph )$ attains its maximum value at $\aleph =1$. So, we get

$$\begin{array}{ccc}\hfill {\Psi }_1\left(\zeta ,ϵ,\aleph \right)& \le & {\Psi }_1\left(\zeta ,ϵ,1\right)={\varsigma }_1(\zeta ,ϵ)+{\varsigma }_2(\zeta ,ϵ)+{\varsigma }_3(\zeta ,ϵ)\hfill \\ & =& 125{\zeta }^6-5400{\zeta }^5+58320{\zeta }^4+5400{\zeta }^3-116640{\zeta }^2+58320+(3888{\zeta }^6-7776{\zeta }^5-11664{\zeta }^4+15552{\zeta }^3\hfill \\ & +& 11664{\zeta }^2-7776\zeta -3888){ϵ}^4+(-2736{\zeta }^6-1296{\zeta }^5+9504{\zeta }^4+2592{\zeta }^3-10800{\zeta }^2-1296\zeta +4032){ϵ}^3\hfill \\ & +& (3348{\zeta }^6+13176{\zeta }^5-66528{\zeta }^4-20952{\zeta }^3+117612{\zeta }^2+7776\zeta -54432){ϵ}^2+(-3300{\zeta }^6+1296{\zeta }^5\hfill \\ & +& 3300{\zeta }^4-2592{\zeta }^3+1296\zeta )ϵ={{\rm Y}}_2(\zeta ,ϵ).\hfill \end{array}$$

The optimal points of ${{\rm Y}}_2(\zeta ,ϵ)$ satisfy the conditions

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {{\rm Y}}_2}{\partial \zeta }}=750{\zeta }^5-27000{\zeta }^4+233280{\zeta }^3+16200{\zeta }^2-233280\zeta +(23328{\zeta }^5-38880{\zeta }^4-46656{\zeta }^3+46656{\zeta }^2+23328\zeta \hfill \\ -7776){ϵ}^4+(-16416{\zeta }^5-6480{\zeta }^4+38016{\zeta }^3+7776{\zeta }^2-21600\zeta -1296){ϵ}^3+(20088{\zeta }^5+65880{\zeta }^4-266112{\zeta }^3\hfill \\ -62856{\zeta }^2+235224\zeta +7776){ϵ}^2+(-19800{\zeta }^5+6480{\zeta }^4+13200{\zeta }^3-7776{\zeta }^2+1296)ϵ=0\hfill \\ {\displaystyle \frac{\partial {{\rm Y}}_2}{\partial ϵ}}=1296\zeta -2592{\zeta }^3+3300{\zeta }^4+1296{\zeta }^5-3300{\zeta }^6+(15552{\zeta }^6-31104{\zeta }^5-46656{\zeta }^4+62208{\zeta }^3+46656{\zeta }^2\hfill \\ -31104\zeta -15552){ϵ}^3+(-8208{\zeta }^6-3888{\zeta }^5+28512{\zeta }^4+7776{\zeta }^3-32400{\zeta }^2-3888\zeta +12096){ϵ}^2+(6696{\zeta }^6\hfill \\ +26352{\zeta }^5-133056{\zeta }^4-41904{\zeta }^3+235224{\zeta }^2+15552\zeta -108864)ϵ=0\hfill \end{array}\right.$$

By calculating, we get

$\left\{\begin{array}{c}{\zeta }_{1,1}\approx 1,\hfill \\ {ϵ}_{1,1}\approx -1.5227,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,2}\approx -1,\hfill \\ {ϵ}_{1,2}\approx 0.5943,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,3}\approx -1,\hfill \\ {ϵ}_{1,3}\approx -0.8998,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,4}=0,\hfill \\ {ϵ}_{1,4}=0,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,5}\approx 1.0010,\hfill \\ {ϵ}_{1,5}\approx -3.5013,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,6}\approx 1.0222,\hfill \\ {ϵ}_{1,6}\approx -0.7916,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,7}\approx 5.0221,\hfill \\ {ϵ}_{1,7}\approx 0.5983,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,8}\approx -0.8746,\hfill \\ {ϵ}_{1,8}\approx 1.2099,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,9}\approx -0.9142,\hfill \\ {ϵ}_{1,9}\approx -1.1388,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,10}\approx -0.9767,\hfill \\ {ϵ}_{1,10}\approx -0.1972,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,11}\approx -1.0001,\hfill \\ {ϵ}_{1,11}\approx -54.4039,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,12}\approx -1.3198,\hfill \\ {ϵ}_{1,12}\approx -1.8163,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,13}\approx -2.0113,\hfill \\ {ϵ}_{1,13}\approx 1.3119,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,14}\approx -6.2159,\hfill \\ {ϵ}_{1,14}\approx 0.7365.\hfill \end{array}\right.$

Thus, there is no optimal point in $(0,1)\times (0,1).$

(1) For $\zeta =0$

$$\begin{array}{c}\hfill {{\rm Y}}_2(0,ϵ)=58320-54432{ϵ}^2+4032{ϵ}^3-3888{ϵ}^4\le 58320.\end{array}$$

(2) For $\zeta =1,$

$$\begin{array}{c}\hfill {{\rm Y}}_2(1,ϵ)=125.\end{array}$$

(3)For $ϵ=0,$

$$\begin{array}{c}\hfill {{\rm Y}}_2(\zeta ,0)=58320-116640{\zeta }^2+5400{\zeta }^3+58320{\zeta }^4-5400{\zeta }^5+125{\zeta }^6\le 58320.\end{array}$$

(4) For $ϵ=1,$

$$\begin{array}{c}\hfill {{\rm Y}}_2(\zeta ,1)=1325{\zeta }^6-7068{\zeta }^4+1836{\zeta }^2+4032={\iota }_1\left(\zeta \right)\le {\iota }_1\left(0.3658\right)=7903.8.\end{array}$$

Thus, we yield

$$\begin{array}{c}\hfill |{\mathcal{H}}_{3,1}\left(\kappa \right)|\le {\displaystyle \frac{58320}{8398080}}={\displaystyle \frac{1}{144}}.\end{array}$$

Theorem 2.5If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\alpha }_4-{\alpha }_2{\alpha }_3|\le {\displaystyle \frac{1}{12}}.$$

The result is sharp for the function ${\kappa }_3\left(\xi \right).$

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (24), (25), (26) and Lemma 1.3, we have

$$\begin{array}{c}\hfill |{\alpha }_4-{\alpha }_2{\alpha }_3|={\displaystyle \frac{1}{12}}|{\beta }_3+{\displaystyle \frac{5}{18}}{\beta }_1{\beta }_2+{\displaystyle \frac{5}{108}}{\beta }_1^3|\le {\displaystyle \frac{1}{12}}.\end{array}$$

(30)

Theorem 2.6If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\alpha }_5-{\alpha }_2{\alpha }_4|\le {\displaystyle \frac{1}{15}}.$$

The result is sharp for the function ${\kappa }_4\left(\xi \right).$

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (24), (26), (27), Lemma 1.4 and 1.5, we have

$$\begin{array}{cc}\hfill |{\alpha }_5-{\alpha }_2{\alpha }_4|& ={\displaystyle \frac{1}{38880}}|2592{\beta }_4+756{\beta }_1{\beta }_3+648{\beta }_2^2+522{\beta }_2{\beta }_1^2-100{\beta }_1^4|\hfill \\ \hfill & ={\displaystyle \frac{1}{38880}}|378({\beta }_1^4+3{\beta }_2{\beta }_1^2+2{\beta }_1{\beta }_3+{\beta }_2^2+{\beta }_4)+2214{\beta }_4-612{\beta }_1^2{\beta }_2+270{\beta }_2^2-478{\beta }_1^4|\hfill \\ \hfill & \le {\displaystyle \frac{1}{38880}}[378+2214(1-|{\beta }_1{|}^2-|{\beta }_2{|}^2)+612|{\beta }_1{|}^2(1-|{\beta }_1{|}^2)+270|{\beta }_2{|}^2+478|{\beta }_1{|}^4]\hfill \\ \hfill & ={\displaystyle \frac{1}{38880}}[2592-1602|{\beta }_1{|}^2-1944|{\beta }_2{|}^2-134|{\beta }_1{|}^4]\le {\displaystyle \frac{2592}{38880}}={\displaystyle \frac{1}{15}}.\hfill \end{array}$$

Theorem 2.7If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\alpha }_5-{\alpha }_3^2|\le {\displaystyle \frac{1}{15}}.$$

The result is sharp for the function ${\kappa }_4\left(\xi \right).$

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (25), (27), Lemma 1.5 and 1.4, we yield

$$\begin{array}{cc}\hfill |{\alpha }_5-{\alpha }_3^2|& ={\displaystyle \frac{1}{12960}}|864{\beta }_4+432{\beta }_1{\beta }_3+56{\beta }_2^2+184{\beta }_2{\beta }_1^2-25{\beta }_1^4|\hfill \\ \hfill & ={\displaystyle \frac{1}{12960}}|216(3{\beta }_2{\beta }_1^2+2{\beta }_1{\beta }_3+{\beta }_2^2+{\beta }_1^4+{\beta }_4)+648{\beta }_4-464{\beta }_1^2{\beta }_2-160{\beta }_2^2-241{\beta }_1^4|\hfill \\ \hfill & \le {\displaystyle \frac{1}{12960}}[216+648(1-|{\beta }_2{|}^2|-|{\beta }_1{|}^2)+464(1-|{\beta }_1{|}^2\left)\right|{\beta }_1{|}^2+160|{\beta }_2{|}^2+241|{\beta }_1{|}^4]\hfill \\ \hfill & ={\displaystyle \frac{1}{12960}}[864-184|{\beta }_1{|}^2-488|{\beta }_2{|}^2-223|{\beta }_1{|}^4]\le {\displaystyle \frac{864}{12960}}={\displaystyle \frac{1}{15}}.\hfill \end{array}$$

Secondly, we study the Hankel determinants of logarithmic coefficients for the class.

Theorem 2.8If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\vartheta }_1|\le {\displaystyle \frac{1}{12}},|{\vartheta }_2|\le {\displaystyle \frac{1}{18}},|{\vartheta }_3|\le {\displaystyle \frac{1}{24}},\left|{\vartheta }_4\right|\le {\displaystyle \frac{1}{30}}.$$

These bounds are sharp for the functions ${\kappa }_1\left(\xi \right),{\kappa }_2\left(\xi \right),{\kappa }_3\left(\xi \right)$ and ${\kappa }_4\left(\xi \right)$, respectively.

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (6), (24), (25), (26) and (27), we achieve

$$\begin{array}{c}\hfill {\vartheta }_1={\displaystyle \frac{1}{12}}{\beta }_1,\end{array}$$

(31)

$$\begin{array}{c}\hfill {\vartheta }_2={\displaystyle \frac{1}{18}}{\beta }_2+{\displaystyle \frac{1}{144}}{\beta }_1^2,\end{array}$$

(32)

$$\begin{array}{c}\hfill {\vartheta }_3={\displaystyle \frac{1}{24}}{\beta }_3+{\displaystyle \frac{5}{432}}{\beta }_1{\beta }_2+{\displaystyle \frac{7}{2592}}{\beta }_1^3,\end{array}$$

(33)

$$\begin{array}{c}\hfill {\vartheta }_4={\displaystyle \frac{1}{155520}}(5184{\beta }_4+1512{\beta }_1{\beta }_3+1044{\beta }_1^2{\beta }_2+816{\beta }_2^2-185{\beta }_1^4).\end{array}$$

(34)

From Lemma 1.1, Lemma 1.2, (31) and (32), we achieve

$$|{\vartheta }_1|\le {\displaystyle \frac{1}{12}},\left|{\vartheta }_2\right|\le {\displaystyle \frac{1}{18}}.$$

Rearranging (33), we have

$$\begin{array}{c}\hfill |{\vartheta }_3|={\displaystyle \frac{1}{24}}|{\beta }_3+{\displaystyle \frac{5}{18}}{\beta }_1{\beta }_2+{\displaystyle \frac{7}{108}}{\beta }_1^3|.\end{array}$$

From Lemma 1.3 (with $\nu ={\displaystyle \frac{5}{18}}$ and $\rho ={\displaystyle \frac{7}{108}}$), we obtain $|{\vartheta }_3|\le {\displaystyle \frac{1}{24}}$.

From (34), Lemma 1.4 and 1.5, we have

$$\begin{array}{cc}\hfill |{\vartheta }_4|& ={\displaystyle \frac{1}{155520}}|756({\beta }_4+3{\beta }_2{\beta }_1^2+2{\beta }_3{\beta }_1+{\beta }_1^4+{\beta }_2^2)+4428{\beta }_4-1224{\beta }_2{\beta }_1^2+60{\beta }_2^2-941{\beta }_1^4|\hfill \\ \hfill & \le {\displaystyle \frac{1}{155520}}[756+4428(1-|{\beta }_1{|}^2-|{\beta }_2{|}^2)+1224|{\beta }_1{|}^2(1-|{\beta }_1{|}^2)+60|{\beta }_2{|}^2+941|{\beta }_1{|}^4]\hfill \\ \hfill & ={\displaystyle \frac{1}{155520}}[5184-3204|{\beta }_1{|}^2-4368|{\beta }_2{|}^2-283|{\beta }_1{|}^4]\le {\displaystyle \frac{5184}{155520}}={\displaystyle \frac{1}{30}}.\hfill \end{array}$$

Theorem 2.9If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\vartheta }_1{\vartheta }_3-{\vartheta }_2^2|\le {\displaystyle \frac{1}{324}}.$$

The bound is sharp for the function ${\kappa }_2\left(\xi \right)$.

Proof Let $\kappa \in {\mathcal{R}}_{LE}$. From (31), (32) and (33) we achieve

$$\begin{array}{c}\hfill |{\vartheta }_1{\vartheta }_3-{\vartheta }_2^2|={\displaystyle \frac{1}{62208}}|-192{\beta }_2^2+12{\beta }_1^2{\beta }_2+216{\beta }_1{\beta }_3+11{\beta }_1^4|.\end{array}$$

(35)

From (35) and Lemma 1.6, we have

$$\begin{array}{ccc}\hfill |{\vartheta }_1{\vartheta }_3-{\vartheta }_2^2|& =& {\displaystyle \frac{1}{62208}}[-216(1-|{\zeta }_1{|}^2\left)\right|{\zeta }_1{|}^2{\zeta }_2^2+216(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2){\zeta }_1{\zeta }_3+12(1-|{\zeta }_1{|}^2){\zeta }_1^2{\zeta }_2+11{\zeta }_1^4\hfill \\ & -& 192(1-|{\zeta }_1{|}^2{)}^2{\zeta }_2^2].\hfill \end{array}$$

By putting $\zeta =|{\zeta }_1|,|{\zeta }_2|=ϵ$, and upon taking $|{\zeta }_3|\le 1$, we get

$$\begin{array}{ccc}\hfill |{\vartheta }_1{\vartheta }_3-{\vartheta }_2^2|& \le & {\displaystyle \frac{1}{62208}}[216(1-{\zeta }^2){\zeta }^2{ϵ}^2+216\zeta (1-{ϵ}^2)(1-{\zeta }^2)+12(1-{\zeta }^2){\zeta }^2ϵ+11{\zeta }^4+192{(1-{\zeta }^2)}^2{ϵ}^2]\hfill \\ & =& {{\rm Y}}_3(\zeta ,ϵ).\hfill \end{array}$$

Differentiating ${{\rm Y}}_3(\zeta ,ϵ)$ with respect to $ϵ$, we have

$${\displaystyle \frac{\partial {{\rm Y}}_3}{\partial ϵ}}={\displaystyle \frac{1}{62208}}[12(1-{\zeta }^2){\zeta }^2+48(1-{\zeta }^2)({\zeta }^2-9\zeta +8)ϵ]\ge 0.$$

Therefore, ${{\rm Y}}_3$ is an increasing function on $ϵ\in [0,1]$. Thus, we get

$$\begin{array}{ccc}\hfill |{\vartheta }_1{\vartheta }_3-{\vartheta }_2^2|& \le & {{\rm Y}}_3(\zeta ,1)={\displaystyle \frac{1}{62208}}(192-156{\zeta }^2-25{\zeta }^4)\le {\displaystyle \frac{192}{62208}}={\displaystyle \frac{1}{324}}.\hfill \end{array}$$

Theorem 2.10If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\vartheta }_2{\vartheta }_4-{\vartheta }_3^2|\le {\displaystyle \frac{1}{576}}.$$

This equality is sharp for the function ${\kappa }_3\left(\xi \right)$.

Proof Let $\kappa \in {\mathcal{R}}_{LE}$. From (32), (33) and (34), we yield

$$\begin{array}{ccc}\hfill {\vartheta }_2{\vartheta }_4-{\vartheta }_3^2& =& {\displaystyle \frac{1}{67184640}}(-116640{\beta }_3^2+18504{\beta }_1^2{\beta }_2^2-5508{\beta }_1^4{\beta }_2-1045{\beta }_1^6+124416{\beta }_2{\beta }_4-28512{\beta }_1{\beta }_2{\beta }_3+19584{\beta }_2^3\hfill \\ & +& 15552{\beta }_1^2{\beta }_4-10584{\beta }_1^3{\beta }_3).\hfill \end{array}$$

(36)

From (36) and Lemma 1.6, we yield

$$\begin{array}{ccc}\hfill {\vartheta }_2{\vartheta }_4-{\vartheta }_3^2& =& {\displaystyle \frac{1}{67184640}}[7776{\overline{{\zeta }_1}}^2(1-|{\zeta }_1{|}^2{)}^2{\zeta }_2^4-15552\overline{{\zeta }_1}{\zeta }_2^2(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2{\zeta }_3-116640(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{{|}^2)}^2{\zeta }_3^2\hfill \\ & -& 1045{\zeta }_1^6+15552{\overline{{\zeta }_1}}^2{\zeta }_1^2(1-|{\zeta }_1{|}^2){\zeta }_2^3-31104{\zeta }_1^2\overline{{\zeta }_1}{\zeta }_2(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2){\zeta }_3-15552{\zeta }_1^2\overline{{\zeta }_2}(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2){\zeta }_3^2\hfill \\ & +& 15552(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_3{|}^2){\zeta }_1^2{\zeta }_4-124416(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2|{\zeta }_2{|}^2{\zeta }_3^2+18504{\zeta }_1^2{\zeta }_2^2(1-|{\zeta }_1{{|}^2)}^2\hfill \\ & +& 124416{\zeta }_2(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_3{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2{\zeta }_4+10584{\zeta }_1^3\overline{{\zeta }_1}(1-|{\zeta }_1{|}^2){\zeta }_2^2-5508{\zeta }_1^4(1-|{\zeta }_1{|}^2){\zeta }_2-28512{\zeta }_1{\zeta }_2\hfill \\ & \times & (1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2){\zeta }_3-10584{\zeta }_1^3(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2){\zeta }_3+28512|{\zeta }_1{|}^2{\zeta }_2^3(1-|{\zeta }_1{|}^2{)}^2+19584(1-|{\zeta }_1{{|}^2)}^3{\zeta }_2^3].\hfill \end{array}$$

Thus, we obtain

$${\vartheta }_2{\vartheta }_4-{\vartheta }_3^2={\displaystyle \frac{1}{67184640}}\left[{\tau }_4\left({\zeta }_1,{\zeta }_2\right)+{\tau }_5\left({\zeta }_1,{\zeta }_2\right){\zeta }_3+{\tau }_6\left({\zeta }_1,{\zeta }_2\right){\zeta }_3^2+{ϝ}_2\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right){\zeta }_4\right],$$

where

$$\begin{array}{ccc}\hfill {\tau }_4\left({\zeta }_1,{\zeta }_2\right)& =& \left(1-|{\zeta }_1{|}^2\right)\left[(1-|{\zeta }_1{|}^2\left)\right(18504{\zeta }_1^2{\zeta }_2^2+8928{\zeta }_2^3|{\zeta }_1{|}^2+19584{\zeta }_2^3+7776{\zeta }_2^4{\overline{{\zeta }_1}}^2)+15552{\zeta }_1^2{\overline{{\zeta }_1}}^2{\zeta }_2^3\right.\hfill \\ & & \left.+10584\overline{{\zeta }_1}{\zeta }_1^3{\zeta }_2^2-5508{\zeta }_1^4{\zeta }_2\right]-1045{\zeta }_1^6,\hfill \\ \hfill {\tau }_5\left({\zeta }_1,{\zeta }_2\right)& =& (1-|{\zeta }_1{|}^2)\left(1-|{\zeta }_2{|}^2\right)\left[(-28512{\zeta }_1{\zeta }_2-15552{\zeta }_2^2\overline{{\zeta }_1})(1-|{\zeta }_1{|}^2)-31104\overline{{\zeta }_1}{\zeta }_1^2{\zeta }_2-10584{\zeta }_1^3\right],\hfill \\ \hfill {\tau }_6\left({\zeta }_1,{\zeta }_2\right)& =& (1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2\left)\right[(-116640-7776|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2)-15552\overline{{\zeta }_2}{\zeta }_1^2],\hfill \\ \hfill {ϝ}_2\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)& =& (1-|{\zeta }_3{|}^2\left)\right(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2\left)\right[15552{\zeta }_1^2+124416(1-|{\zeta }_1{|}^2\left){\zeta }_2\right]\hfill \end{array}$$

By setting $\zeta =|{\zeta }_1|,|{\zeta }_2|=ϵ,\aleph =|{\zeta }_3|$ and upon taking $|{\zeta }_4|\le 1$, we have

$$\begin{array}{ccc}\hfill \left|{\vartheta }_2{\vartheta }_4-{\vartheta }_3^2\right|& \le & {\displaystyle \frac{1}{67184640}}\left(\left|{\tau }_4\left({\zeta }_1,{\zeta }_2\right)\right|+\left|{\tau }_5\left({\zeta }_1,{\zeta }_2\right)\right|\aleph +\left|{\tau }_6\left({\zeta }_1,{\zeta }_2\right)\right|{\aleph }^2+\left|{ϝ}_2\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)\right|\right)\hfill \\ & \le & {\displaystyle \frac{1}{67184640}}{\Psi }_2\left(\zeta ,ϵ,\aleph \right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\Psi }_2\left(\zeta ,ϵ,\aleph \right)={\varsigma }_5\left(\zeta ,ϵ\right)+{\varsigma }_6\left(\zeta ,ϵ\right)\aleph +{\varsigma }_7\left(\zeta ,ϵ\right){\aleph }^2+{\varsigma }_8\left(\zeta ,ϵ\right)(1-{\aleph }^2),\end{array}$$

with

$$\begin{array}{ccc}\hfill {\varsigma }_5\left(\zeta ,ϵ\right)& =& \left(1-{\zeta }^2\right)\left[(7776{ϵ}^4{\zeta }^2+8928{ϵ}^3{\zeta }^2+19584{ϵ}^3+18504{ϵ}^2{\zeta }^2)(1-{\zeta }^2)+15552{ϵ}^3{\zeta }^4+10584{ϵ}^2{\zeta }^4\right.\hfill \\ & & \left.+5508{\zeta }^4ϵ\right]+1045{\zeta }^6,\hfill \\ \hfill {\varsigma }_6\left(\zeta ,ϵ\right)& =& \left(1-{ϵ}^2\right)(1-{\zeta }^2)\left[(28512ϵ\zeta +15552{ϵ}^2\zeta )(1-{\zeta }^2)+10584{\zeta }^3+31104{\zeta }^3ϵ\right],\hfill \\ \hfill {\varsigma }_7\left(\zeta ,ϵ\right)& =& \left(1-{\zeta }^2\right)(1-{ϵ}^2)\left[15552{\zeta }^2ϵ+(7776{ϵ}^2+116640)(1-{\zeta }^2)\right],\hfill \\ \hfill {\varsigma }_8\left(\zeta ,ϵ\right)& =& (1-{ϵ}^2)(1-{\zeta }^2)\left[15552{\zeta }^2+124416ϵ(1-{\zeta }^2)\right].\hfill \end{array}$$

Note that

$$\begin{array}{ccc}\hfill {\varsigma }_7(\zeta ,ϵ))& \le & (1-{ϵ}^2)(1-{\zeta }^2)[(116640+7776{ϵ}^2)(1-{\zeta }^2)+15552{\zeta }^2]={\varrho }_1(\zeta ,ϵ),\hfill \end{array}$$

Thus, we receive

$${\Psi }_2\left(\zeta ,ϵ,\aleph \right)\le {\varsigma }_5\left(\zeta ,ϵ\right)+{\varsigma }_6\left(\zeta ,ϵ\right)\aleph +{\varrho }_1\left(\zeta ,ϵ\right){\aleph }^2+{\varsigma }_8\left(\zeta ,ϵ\right)(1-{\aleph }^2)={\Gamma }_1(\zeta ,ϵ,\aleph ).$$

Differentiating ${\Gamma }_1(\zeta ,ϵ,\aleph )$ with respect to ℵ, we yield

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\Gamma }_1}{\partial \aleph }}& =& {\varsigma }_6(\zeta ,ϵ)+2\aleph [{\varrho }_1(\zeta ,ϵ)-{\varsigma }_8(\zeta ,ϵ)]\hfill \\ & & ={\varsigma }_6(\zeta ,ϵ)+15552\aleph {(1-{\zeta }^2)}^2(15-16ϵ+{ϵ}^2)(1-{ϵ}^2)\ge 0.\hfill \end{array}$$

Obviously, ${\Gamma }_1(\zeta ,ϵ,\aleph )$ attains its maximum value at $\aleph =1$. So, we have

$$\begin{array}{ccc}\hfill {\Psi }_2\left(\zeta ,ϵ,\aleph \right)& \le & {\Gamma }_1\left(\zeta ,ϵ,\aleph \right)\le {\Gamma }_1\left(\zeta ,ϵ,1\right)={\varsigma }_5(\zeta ,ϵ)+{\varsigma }_6(\zeta ,ϵ)+{\varrho }_1(\zeta ,ϵ)\hfill \\ & =& 116640-217728{\zeta }^2+10584{\zeta }^3+101088{\zeta }^4-10584{\zeta }^5+1045{\zeta }^6+(-7776-15552\zeta +23328{\zeta }^2\hfill \\ & +& 31104{\zeta }^3-23328{\zeta }^4-15552{\zeta }^5+7776{\zeta }^6){ϵ}^4+(19584-28512\zeta -30240{\zeta }^2+25920{\zeta }^3+17280{\zeta }^4\hfill \\ & +& 2592{\zeta }^5-6624{\zeta }^6){ϵ}^3+(-108864+15552\zeta +220680{\zeta }^2-41688{\zeta }^3-119736{\zeta }^4+26136{\zeta }^5+7920{\zeta }^6){ϵ}^2\hfill \\ & +& (28512\zeta -25920{\zeta }^3+5508{\zeta }^4-2592{\zeta }^5-5508{\zeta }^6)ϵ={{\rm Y}}_4(\zeta ,ϵ).\hfill \end{array}$$

Consider

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {{\rm Y}}_4}{\partial \zeta }}=-435456\zeta +31752{\zeta }^2+404352{\zeta }^3-52920{\zeta }^4+6270{\zeta }^5+(-15552+46656\zeta +93312{\zeta }^2-93312{\zeta }^3-77760{\zeta }^4\hfill \\ +46656{\zeta }^5){ϵ}^4+(-28512-60480\zeta +77760{\zeta }^2+69120{\zeta }^3+12960{\zeta }^4-39744{\zeta }^5){ϵ}^3+(15552+441360\zeta -125064{\zeta }^2\hfill \\ -478944{\zeta }^3+130680{\zeta }^4+47520{\zeta }^5){ϵ}^2+(28512-77760{\zeta }^2+22032{\zeta }^3-12960{\zeta }^4-33048{\zeta }^5)ϵ=0\hfill \\ {\displaystyle \frac{\partial {{\rm Y}}_4}{\partial ϵ}}=-5508{\zeta }^6-2592{\zeta }^5+5508{\zeta }^4-25920{\zeta }^3+28512\zeta +(-31104-62208\zeta +93312{\zeta }^2+124416{\zeta }^3-93312{\zeta }^4\hfill \\ -62208{\zeta }^5+31104{\zeta }^6){ϵ}^3+(58752-85536\zeta -90720{\zeta }^2+77760{\zeta }^3+51840{\zeta }^4+7776{\zeta }^5-19872{\zeta }^6){ϵ}^2+(-217728\hfill \\ +31104\zeta +441360{\zeta }^2-83376{\zeta }^3-239472{\zeta }^4+52272{\zeta }^5+15840{\zeta }^6)ϵ=0\hfill \end{array}\right.$$

By calculating, we achieve

$\left\{\begin{array}{c}{\zeta }_{1,1}\approx 1,\hfill \\ {ϵ}_{1,1}\approx 1.4111,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,2}\approx 1,\hfill \\ {ϵ}_{1,2}\approx -1.8421,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,3}\approx 1,\hfill \\ {ϵ}_{1,3}\approx -0.5690,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,4}\approx -1,\hfill \\ {ϵ}_{1,4}\approx 0.6405,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,5}\approx -1,\hfill \\ {ϵ}_{1,5}\approx -0.8343,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,6}\approx -1,\hfill \\ {ϵ}_{1,6}\approx 0.0735,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,7}=0,\hfill \\ {ϵ}_{1,7}=0,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,8}\approx 7.5637,\hfill \\ {ϵ}_{1,8}\approx 0.3915,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,9}\approx -0.6852,\hfill \\ {ϵ}_{1,9}\approx 1.1077,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,10}\approx -0.7937,\hfill \\ {ϵ}_{1,10}\approx -9.0663,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,11}\approx -0.9888,\hfill \\ {ϵ}_{1,11}\approx 0.4081,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,12}\approx -1.0287,\hfill \\ {ϵ}_{1,12}\approx -0.5214,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,13}\approx -1.4943,\hfill \\ {ϵ}_{1,13}\approx 2.3324,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,14}\approx -1.8490,\hfill \\ {ϵ}_{1,14}\approx 1.6682,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,15}\approx -57.6080,\hfill \\ {ϵ}_{1,15}\approx 0.4434.\hfill \end{array}\right.$

Thus, there is no optimal point in $(0,1)\times (0,1).$

(5) For $\zeta =0,$

$$\begin{array}{c}\hfill {{\rm Y}}_4(0,ϵ)=116640-108864{ϵ}^2+19584{ϵ}^3-7776{ϵ}^4\le 116640.\end{array}$$

(6) For $\zeta =1,$

$$\begin{array}{c}\hfill {{\rm Y}}_4(1,ϵ)=1045.\end{array}$$

(7) For $ϵ=0,$

$$\begin{array}{c}\hfill {{\rm Y}}_4(\zeta ,0)=116640-217728{\zeta }^2+10584{\zeta }^3+101088{\zeta }^4-10584{\zeta }^5+1045{\zeta }^6\le 116640.\end{array}$$

(8) For $ϵ=1,$

$$\begin{array}{c}\hfill {{\rm Y}}_4(\zeta ,1)=4609{\zeta }^6-19188{\zeta }^4-3960{\zeta }^2+19584\le 19584.\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill |{\vartheta }_2{\vartheta }_4-{\vartheta }_3^2|\le {\displaystyle \frac{116640}{67184640}}={\displaystyle \frac{1}{576}}.\end{array}$$

Finally, we evaluate the higher-order Schwarzian derivative for ${\mathcal{R}}_{LE}.$

Theorem 2.11If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|S_3|\le {\displaystyle \frac{2}{3}},|S_4|\le 2,|S_5|\le 8.$$

These equalities are sharp for the functions ${\kappa }_2\left(\xi \right),{\kappa }_3\left(\xi \right)$ and ${\kappa }_4\left(\xi \right)$.

Proof Let $\kappa \in {\mathcal{R}}_{LE}$. From (13), (24), (25), (26) and (27), we get

$$\begin{array}{c}\hfill S_3={\displaystyle \frac{2}{3}}{\beta }_2,\end{array}$$

(37)

$$\begin{array}{c}\hfill S_4=2({\beta }_3-{\displaystyle \frac{1}{6}}{\beta }_1{\beta }_2+{\displaystyle \frac{5}{108}}{\beta }_1^3),\end{array}$$

(38)

$$\begin{array}{c}\hfill S_5={\displaystyle \frac{1}{324}}(2592{\beta }_4-864{\beta }_1{\beta }_3+432{\beta }_1^2{\beta }_2-216{\beta }_2^2-175{\beta }_1^4).\end{array}$$

(39)

From (37), (38), Lemma 1.1 and 1.3, we receive

$$|S_3|\le {\displaystyle \frac{2}{3}},\left|S_4\right|\le 2.$$

From (39), Lemma 1.4 and 1.5, we receive

$$\begin{array}{cc}\hfill |S_5|& ={\displaystyle \frac{1}{324}}|432({\beta }_4-{\beta }_2^2+3{\beta }_2{\beta }_1^2-2{\beta }_3{\beta }_1-{\beta }_1^4)+2160{\beta }_4+216{\beta }_2^2-864{\beta }_2{\beta }_1^2+257{\beta }_1^4|\hfill \\ \hfill & \le {\displaystyle \frac{1}{324}}[432+2160(1-|{\beta }_1{|}^2-|{\beta }_2{|}^2)+216|{\beta }_2{|}^2+864(1-|{\beta }_1{|}^2\left)\right|{\beta }_1{|}^2+257|{\beta }_1{|}^4]\hfill \\ \hfill & ={\displaystyle \frac{1}{324}}[2592-1296|{\beta }_1{|}^2-1944|{\beta }_2{|}^2-607|{\beta }_1{|}^4]\le {\displaystyle \frac{2592}{324}}=8.\hfill \end{array}$$

Theorem 2.12If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|S_3{S}_5-S_4^2|\le 4.$$

These equality is sharp for the function ${\kappa }_3\left(\xi \right)$.

Proof Let $\kappa \in {\mathcal{R}}_{LE}$. From (37), (38) and (39), we receive

$$\begin{array}{ccc}\hfill S_3{S}_5-S_4^2& =& {\displaystyle \frac{1}{2916}}[-11664{\beta }_3^2+180{\beta }_1^4{\beta }_2+2268{\beta }_1^2{\beta }_2^2-1075{\beta }_1^6+15552{\beta }_2{\beta }_4-1296{\beta }_1{\beta }_2{\beta }_3\hfill \\ & -& 1080{\beta }_1^3{\beta }_3-1296{\beta }_2^3].\hfill \end{array}$$

(40)

From Lemma 1.6 and (40), we have

$$\begin{array}{ccc}\hfill S_3{S}_5-S_4^2& =& {\displaystyle \frac{1}{2916}}[-1296{\zeta }_2^3(1-|{\zeta }_1{|}^2{)}^3+180{\zeta }_2{\zeta }_1^4(1-|{\zeta }_1{|}^2)+2268{\zeta }_2^2{\zeta }_1^2{(1-{\zeta }_1{|}^2)}^2+3888(1-|{\zeta }_1{{|}^2)}^2{\zeta }_2^4{\overline{{\zeta }_1}}^2\hfill \\ & -& 15552{\zeta }_3^2|{\zeta }_2{|}^2(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2-1075{\zeta }_1^6-7776{\zeta }_3{\zeta }_2^2\overline{{\zeta }_1}(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2)+15552{\zeta }_4{\zeta }_2(1-|{\zeta }_2{|}^2)\hfill \\ & \times & (1-|{\zeta }_3{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2+1296|{\zeta }_1{|}^2{\zeta }_2^3(1-|{\zeta }_1{|}^2{)}^2-1296{\zeta }_3{\zeta }_2{\zeta }_1(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2)-11664{\zeta }_3^2(1-|{\zeta }_1{{|}^2)}^2\hfill \\ & \times & (1-|{\zeta }_2{|}^2{)}^2+1080{\zeta }_2^2\overline{{\zeta }_1}{\zeta }_1^3(1-|{\zeta }_1{|}^2)-1080{\zeta }_3{\zeta }_1^3(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2)].\hfill \end{array}$$

Therefore, we have

$$S_3{S}_5-S_4^2={\displaystyle \frac{1}{2916}}\left[{\tau }_7\left({\zeta }_1,{\zeta }_2\right)+{\tau }_8\left({\zeta }_1,{\zeta }_2\right){\zeta }_3+{\tau }_9\left({\zeta }_1,{\zeta }_2\right){\zeta }_3^2+{ϝ}_3\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right){\zeta }_4\right],$$

where

$$\begin{array}{ccc}\hfill {\tau }_7\left({\zeta }_1,{\zeta }_2\right)& =& \left(1-|{\zeta }_1{|}^2\right)\left\{[2268{\zeta }_1^2{\zeta }_2^2+2592|{\zeta }_1{|}^2{\zeta }_2^3-1296{\zeta }_2^3+3888{\overline{{\zeta }_1}}^2{\zeta }_2^4\left]\right(1-|{\zeta }_1{|}^2)+1080\overline{{\zeta }_1}{\zeta }_1^3{\zeta }_2^2\right.\hfill \\ & & \left.+180{\zeta }_2{\zeta }_1^4\right\}-1075{\zeta }_1^6,\hfill \\ \hfill {\tau }_8\left({\zeta }_1,{\zeta }_2\right)& =& (1-|{\zeta }_1{|}^2)\left(1-|{\zeta }_2{|}^2\right)\left[(-1296{\zeta }_1{\zeta }_2-7776\overline{{\zeta }_1}{\zeta }_2^2)(1-|{\zeta }_1{|}^2)-1080{\zeta }_1^3\right],\hfill \\ \hfill {\tau }_9\left({\zeta }_1,{\zeta }_2\right)& =& -3888(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2\left)\right(3+|{\zeta }_2{|}^2),\hfill \\ \hfill {ϝ}_3\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)& =& 15552{\zeta }_2(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_3{|}^2\left)\right(1-|{\zeta }_1{{|}^2)}^2.\hfill \end{array}$$

By putting $\aleph =|{\zeta }_3|,ϵ=|{\zeta }_2|,\zeta =|{\zeta }_1|$ and upon taking $|{\zeta }_4|\le 1$, we receive

$$\begin{array}{ccc}\hfill S_3{S}_5-S_4^2& \le & {\displaystyle \frac{1}{2361960}}\left(\left|{\tau }_7\left({\zeta }_1,{\zeta }_2\right)\right|+\left|{\tau }_8\left({\zeta }_1,{\zeta }_2\right)\right|\aleph +\left|{\tau }_9\left({\zeta }_1,{\zeta }_2\right)\right|{\aleph }^2+\left|{ϝ}_3\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)\right|\right)\hfill \\ & \le & {\displaystyle \frac{1}{2916}}{\Psi }_3\left(\zeta ,ϵ,\aleph \right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\Psi }_3\left(\zeta ,ϵ,\aleph \right)={\varsigma }_9\left(\zeta ,ϵ\right)+{\varsigma }_{10}\left(\zeta ,ϵ\right)\aleph +{\varsigma }_{11}\left(\zeta ,ϵ\right){\aleph }^2+{\varsigma }_{12}\left(\zeta ,ϵ\right)(1-{\aleph }^2),\end{array}$$

with

$$\begin{array}{ccc}\hfill {\varsigma }_9\left(\zeta ,ϵ\right)& =& \left(1-{\zeta }^2\right)\left[(3888{\zeta }^2{ϵ}^4+1296{ϵ}^3+2592{ϵ}^3{\zeta }^2+2268{\zeta }^2{ϵ}^2)(1-{\zeta }^2)+1080{ϵ}^2{\zeta }^4+180{\zeta }^4ϵ\right]+1075{\zeta }^6,\hfill \\ \hfill {\varsigma }_{10}\left(\zeta ,ϵ\right)& =& \left(1-{ϵ}^2\right)(1-{\zeta }^2)\left[(1296\zeta ϵ+7776{ϵ}^2\zeta )(1-{\zeta }^2)+1080{\zeta }^3\right],\hfill \\ \hfill {\varsigma }_{11}\left(\zeta ,ϵ\right)& =& 3888{\left(1-{\zeta }^2\right)}^2(1-{ϵ}^2)({ϵ}^2+3),\hfill \\ \hfill {\varsigma }_{12}\left(\zeta ,ϵ\right)& =& 15552{\left(1-{\zeta }^2\right)}^2(ϵ-{ϵ}^3).\hfill \end{array}$$

Differentiating ${\Psi }_3(\zeta ,ϵ,\aleph )$ with respect to ℵ, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\Psi }_3}{\partial \aleph }}& =& {\varsigma }_{10}(\zeta ,ϵ)+2[{\varsigma }_{11}(\zeta ,ϵ)-{\varsigma }_{12}(\zeta ,ϵ)]\aleph \hfill \\ & & ={\varsigma }_{10}(\zeta ,ϵ)+7776(1-{ϵ}^2)({ϵ}^2-4ϵ+3){(1-{\zeta }^2)}^2\aleph \ge 0.\hfill \end{array}$$

Thus, ${\Psi }_3(\zeta ,ϵ,\aleph )$ attains its maximum value at $\aleph =1$. Therefore, we have

$$\begin{array}{ccc}\hfill {\Psi }_3\left(\zeta ,ϵ,\aleph \right)& \le & {\Psi }_3\left(\zeta ,ϵ,1\right)={\varsigma }_9(\zeta ,ϵ)+{\varsigma }_{10}(\zeta ,ϵ)+{\varsigma }_{11}(\zeta ,ϵ)\hfill \\ & =& 11664-23328{\zeta }^2+1080{\zeta }^3+11664{\zeta }^4-1080{\zeta }^5+1075{\zeta }^6+(-3888-7776\zeta +11664{\zeta }^2+15552{\zeta }^3\hfill \\ & -& 11664{\zeta }^4-7776{\zeta }^5+3888{\zeta }^6){ϵ}^4+(1296-1296\zeta +2592{\zeta }^3-3888{\zeta }^4-1296{\zeta }^5+2592{\zeta }^6){ϵ}^3+(-7776\hfill \\ & +& 7776\zeta +17820{\zeta }^2-16632{\zeta }^3-11232{\zeta }^4+8856{\zeta }^5+1188{\zeta }^6){ϵ}^2+(1296\zeta -2592{\zeta }^3+180{\zeta }^4+1296{\zeta }^5\hfill \\ & -& 180{\zeta }^6)ϵ={{\rm Y}}_5(\zeta ,ϵ).\hfill \end{array}$$

Consider

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {{\rm Y}}_5}{\partial \zeta }}=-46656\zeta +3240{\zeta }^2+46656{\zeta }^3-5400{\zeta }^4+6450{\zeta }^5+(-7776+23328\zeta +46656{\zeta }^2-46656{\zeta }^3-38880{\zeta }^4\hfill \\ +23328{\zeta }^5){ϵ}^4+(-1296+7776{\zeta }^2-15552{\zeta }^3-6480{\zeta }^4+15552{\zeta }^5){ϵ}^3+(7776+35640\zeta -49896{\zeta }^2-44928{\zeta }^3\hfill \\ +44280{\zeta }^4+7128{\zeta }^5){ϵ}^2+(1296-7776{\zeta }^2+720{\zeta }^3+6480{\zeta }^4-1080{\zeta }^5)ϵ=0\hfill \\ {\displaystyle \frac{\partial {{\rm Y}}_5}{\partial ϵ}}=-180{\zeta }^6+1296{\zeta }^5+180{\zeta }^4-2592{\zeta }^3+1296\zeta +(-15552-31104\zeta +46656{\zeta }^2+62208{\zeta }^3-46656{\zeta }^4\hfill \\ -31104{\zeta }^5+15552{\zeta }^6){ϵ}^3+(3888-3888\zeta +7776{\zeta }^3-11664{\zeta }^4-3888{\zeta }^5+7776{\zeta }^6){ϵ}^2+(-15552+15552\zeta \hfill \\ +35640{\zeta }^2-33264{\zeta }^3-22464{\zeta }^4+17712{\zeta }^5+2376{\zeta }^6)ϵ=0\hfill \end{array}\right.$$

By calculating, we achieve

$\left\{\begin{array}{c}{\zeta }_{1,1}\approx 1,\hfill \\ {ϵ}_{1,1}\approx 11.9167,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,2}\approx -1,\hfill \\ {ϵ}_{1,2}\approx 1.3707,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,3}\approx -1,\hfill \\ {ϵ}_{1,3}\approx -1.4540,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,4}=0,\hfill \\ {ϵ}_{1,4}=0,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,5}\approx 0.9618,\hfill \\ {ϵ}_{1,5}\approx 0.6338,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,6}\approx 1,\hfill \\ {ϵ}_{1,6}\approx 23.7441,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,7}\approx -0.6390,\hfill \\ {ϵ}_{1,7}\approx 1.0810,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,8}\approx -0.9299,\hfill \\ {ϵ}_{1,8}\approx 0.1489,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,9}\approx -0.9312,\hfill \\ {ϵ}_{1,9}\approx -0.0994,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,10}\approx -1.0017,\hfill \\ {ϵ}_{1,10}\approx -6.6337.\hfill \end{array}\right.$

Thus, there is a optimal point $({\zeta }_{1,5},{ϵ}_{1,5})$ in $(0,1)\times (0,1).$ Therefore, we have $max{{\rm Y}}_5(\zeta ,ϵ)={{\rm Y}}_5(0.9618,0.6338)=999.659432.$

(9) For $\zeta =0,$

$$\begin{array}{c}\hfill {{\rm Y}}_5(0,ϵ)=11664-7776{ϵ}^2+1296{ϵ}^3-3888{ϵ}^4\le 11664.\end{array}$$

(10) For $\zeta =1,$

$$\begin{array}{c}\hfill {{\rm Y}}_5(1,ϵ)=1075.\end{array}$$

(11) For $ϵ=0,$

$$\begin{array}{c}\hfill {{\rm Y}}_5(\zeta ,0)=11664-23328{\zeta }^2+1080{\zeta }^3+11664{\zeta }^4-1080{\zeta }^5+1075{\zeta }^6\le 11664.\end{array}$$

(12) For $ϵ=1,$

$$\begin{array}{c}\hfill {{\rm Y}}_5(\zeta ,1)=8563{\zeta }^6-14940{\zeta }^4+6156{\zeta }^2+1296={\iota }_2\left(\zeta \right)\le {\iota }_2\left(0.5173\right)=2037.6.\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill |S_3{S}_5-S_4^2|\le {\displaystyle \frac{11664}{2916}}=4.\end{array}$$

3. The Sharp Functional Inequalities of Inverse Functions for the Class ${\mathcal{R}}_{LE}$

Theorem 3.1If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|A_2|\le {\displaystyle \frac{1}{6}},|A_3|\le {\displaystyle \frac{1}{9}},|A_4|\le {\displaystyle \frac{1}{12}},\left|A_5\right|\le {\displaystyle \frac{1}{15}}.$$

These bounds are sharp for the functions ${\kappa }_1\left(\xi \right),{\kappa }_2\left(\xi \right),{\kappa }_3\left(\xi \right)$ and ${\kappa }_4\left(\xi \right)$, respectively.

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (24), (25), (26), (27) and (8), we receive

$$\begin{array}{c}\hfill A_2=-{\displaystyle \frac{1}{6}}{\beta }_1,\end{array}$$

(41)

$$\begin{array}{c}\hfill A_3=-{\displaystyle \frac{1}{9}}{\beta }_2+{\displaystyle \frac{1}{36}}{\beta }_1^2,\end{array}$$

(42)

$$\begin{array}{c}\hfill A_4=-{\displaystyle \frac{1}{12}}{\beta }_3+{\displaystyle \frac{11}{216}}{\beta }_1{\beta }_2-{\displaystyle \frac{11}{1296}}{\beta }_1^3,\end{array}$$

(43)

$$\begin{array}{c}\hfill A_5=-{\displaystyle \frac{1}{12960}}(864{\beta }_4-648{\beta }_1{\beta }_3+324{\beta }_1^2{\beta }_2-264{\beta }_2^2-85{\beta }_1^4).\end{array}$$

(44)

From (41), (42), Lemma 1.1 and 1.2, we achieve

$$|A_2|\le {\displaystyle \frac{1}{6}},\left|A_3\right|\le {\displaystyle \frac{1}{9}}.$$

Rearranging (43), we receive

$$\begin{array}{c}\hfill |A_4|={\displaystyle \frac{1}{12}}|{\beta }_3-{\displaystyle \frac{11}{18}}{\beta }_1{\beta }_2+{\displaystyle \frac{11}{108}}{\beta }_1^3|.\end{array}$$

From Lemma 1.3 (with $\nu =-{\displaystyle \frac{11}{18}}$ and $\rho ={\displaystyle \frac{11}{108}}$), we get ${\displaystyle \frac{4}{27}}{\left(\right|\nu |+1)}^3-\left(\right|\nu |+1)=-{\displaystyle \frac{156136}{157464}}<\rho ={\displaystyle \frac{11}{108}}.$ Then by Lemma 1.3, we have

$$|A_4|\le {\displaystyle \frac{1}{12}}$$

.

From Lemma 1.4, Lemma 1.2, Lemma 1.5 and (44), we yield

$$\begin{array}{cc}\hfill |A_5|& ={\displaystyle \frac{1}{12960}}|324({\beta }_4-{\beta }_1^4-2{\beta }_3{\beta }_1-{\beta }_2^2+3{\beta }_2{\beta }_1^2)+540{\beta }_4+60{\beta }_2^2-239{\beta }_1^2({\beta }_2-{\beta }_1^2)-409{\beta }_1^2{\beta }_2|\hfill \\ \hfill & \le {\displaystyle \frac{1}{12960}}[324+540(1-|{\beta }_1{|}^2-|{\beta }_2{|}^2)+60|{\beta }_2{|}^2+239|{\beta }_1{|}^2+409|{\beta }_1{|}^2|{\beta }_2\left|\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{12960}}[864-301|{\beta }_1{|}^2-480|{\beta }_2{|}^2+409|{\beta }_1{|}^2|{\beta }_2\left|\right].\hfill \end{array}$$

Setting $|{\beta }_2|=y,x=|{\beta }_1|,$ we get

$$\begin{array}{c}\hfill |A_5|\le {\displaystyle \frac{1}{12960}}[864-301x^2-480y^2+409x^{2}y]={\displaystyle \frac{1}{12960}}{\varpi }_1(x,y).\end{array}$$

Consider

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {\varpi }_1}{\partial x}}=-602x+818xy=0,\hfill \\ {\displaystyle \frac{\partial {\varpi }_2}{\partial y}}=-960y+409x^2=0.\hfill \end{array}\right.$$

By calculating, we achieve

$\left\{\begin{array}{c}{x}_1=0,\hfill \\ y_1=0,\hfill \end{array}\right.$$\left\{\begin{array}{c}{x}_2=1.3143,\hfill \\ y_2=0.7359,\hfill \end{array}\right.$$\left\{\begin{array}{c}{x}_3=-1.3143,\hfill \\ y_3=0.7359.\hfill \end{array}\right.$

Thus, there is no optimal point in $(0,1)\times (0,1).$

(13) For $x=0$

$$\begin{array}{c}\hfill {\varpi }_1(0,y)=864-480y^2\le 864.\end{array}$$

(14) For $y=0,$

$$\begin{array}{c}\hfill {\varpi }_1(x,0)=864-301x^2\le 864\end{array}$$

(15) For $y=1-x^2,$

$$\begin{array}{c}\hfill {\varpi }_1(x,1-x^2)=384+1068x^2-889x^4={\iota }_3\left(x\right)\le {\iota }_3\left(0.7750\right)=704.7604.\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill |A_5|\le {\displaystyle \frac{864}{12960}}={\displaystyle \frac{1}{15}}.\end{array}$$

Using equations (41), (42) and Lemma 1.2, we easily get the following theorem.

Theorem 3.2If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|A_3-\gamma A_2^2|\le {\displaystyle \frac{1}{9}}max\{1,{\displaystyle \frac{|1-\gamma |}{4}}\}.$$

This result is sharp.

Theorem 3.3If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|A_2{A}_4-A_3^2|\le {\displaystyle \frac{1}{81}}.$$

The equality is sharp for the function ${\kappa }_2\left(\xi \right).$

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (41), (42), (43), we have

$$\begin{array}{c}\hfill {\mathcal{H}}_{2,2}\left({\kappa }^{-1}\right)={\displaystyle \frac{1}{7776}}(108{\beta }_3{\beta }_1-96{\beta }_2^2-18{\beta }_1^2{\beta }_2+5{\beta }_1^4).\end{array}$$

(45)

From Lemma 1.6 and (45), we yield

$$\begin{array}{ccc}\hfill \left\}\right\}{H}_{2,2}\left({\kappa }^{-1}\right)& =& {\displaystyle \frac{1}{7776}}[-108{\zeta }_2^2(1-|{\zeta }_1{|}^2\left)\right|{\zeta }_1{|}^2+108{\zeta }_3{\zeta }_1(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2)-18{\zeta }_2{\zeta }_1^2(1-|{\zeta }_1{|}^2)+5{\zeta }_1^4\hfill \\ & -& 96{\zeta }_2^2(1-|{\zeta }_1{|}^2{)}^2].\hfill \end{array}$$

Setting $\zeta =|{\zeta }_1|,|{\zeta }_2|=ϵ$ and upon taking $|{\zeta }_3|\le 1$, we obtain

$$\begin{array}{ccc}\hfill \left|\right\}\}{H}_{2,2}\left({\kappa }^{-1}\right)|& \le & {\displaystyle \frac{1}{7776}}[108{ϵ}^2(1-{\zeta }^2){\zeta }^2+108(1-{ϵ}^2)(1-{\zeta }^2)\zeta +18{\zeta }^2(1-{\zeta }^2)ϵ+5{\zeta }^4+96{ϵ}^2{(1-{\zeta }^2)}^2]\hfill \\ & =& {\displaystyle \frac{1}{7776}}{{\rm Y}}_6(\zeta ,ϵ).\hfill \end{array}$$

Differentiating ${{\rm Y}}_6(\zeta ,ϵ)$ with respect to $ϵ$, we recieve

$${\displaystyle \frac{\partial {{\rm Y}}_6}{\partial ϵ}}=18(1-{\zeta }^2){\zeta }^2+24ϵ(1-{\zeta }^2)(8-9\zeta +{\zeta }^2)\ge 0.$$

Thus, ${{\rm Y}}_6(\zeta ,ϵ)$ attains its maximum value at $ϵ=1$. Therefore, we get

$$\begin{array}{ccc}\hfill |{\mathcal{H}}_{2,2}\left({\kappa }^{-1}\right)|& \le & {\displaystyle \frac{1}{7776}}{{\rm Y}}_6(\zeta ,1)={\displaystyle \frac{1}{7776}}(96-66{\zeta }^2-25{\zeta }^4)\le {\displaystyle \frac{96}{7776}}={\displaystyle \frac{1}{81}}.\hfill \end{array}$$

Theorem 3.4Let $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\mathcal{H}}_{3,1}\left({\tau }^{-1}\right)|\le {\displaystyle \frac{1}{144}}.$$

The equality is best possible for the function ${\kappa }_3\left(\xi \right).$

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (4), (41), (42), (43) and (44), we have

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{3,1}\left({\kappa }^{-1}\right)& =& {\displaystyle \frac{1}{8398080}}(-7488{\beta }_2^3+8748{\beta }_2^2{\beta }_1^2-3300{\beta }_2{\beta }_1^4-125{\beta }_1^6+62208{\beta }_4{\beta }_2-1296{\beta }_3{\beta }_2{\beta }_1\hfill \\ & -& 5400{\beta }_3{\beta }_1^3-58320{\beta }_3^2).\hfill \end{array}$$

(46)

From Lemma 1.6 and (46), we have

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{3,1}\left({\kappa }^{-1}\right)& =& {\displaystyle \frac{1}{8398080}}[-125{\zeta }_1^6+8748{\zeta }_1^2{\zeta }_2^2(1-|{\zeta }_1{|}^2{)}^2-3300{\zeta }_1^4{\zeta }_2(1-|{\zeta }_1{|}^2)+62208(1-|{\zeta }_3{|}^2\left)\right(1-|{\zeta }_1{{|}^2)}^2\hfill \\ & \times & (1-|{\zeta }_2{|}^2){\zeta }_4{\zeta }_2-7488(1-|{\zeta }_1{|}^2{)}^3{\zeta }_2^3-1296{\zeta }_3{\zeta }_1{\zeta }_2(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2+1296|{\zeta }_1{|}^2{\zeta }_2^3(1-|{\zeta }_1{{|}^2)}^2\hfill \\ & -& 5400{\zeta }_1^3{\zeta }_3(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2)+5400{\zeta }_1^3\overline{{\zeta }_1}{\zeta }_2^2(1-|{\zeta }_1{|}^2)-62208{\zeta }_3^2|{\zeta }_2{|}^2(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2)\hfill \\ & -& 58320{\zeta }_3^2(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2{)}^2-7776{\zeta }_3{\zeta }_2^2\overline{{\zeta }_1}(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2+3888{\zeta }_2^4{\overline{{\zeta }_1}}^2(1-|{\zeta }_1{|}^2{)}^2].\hfill \end{array}$$

Thus, we receive

$${\mathcal{H}}_{3,1}\left({\kappa }^{-1}\right)={\displaystyle \frac{1}{8398080}}\left[{\tau }_{10}\left({\zeta }_1,{\zeta }_2\right)+{\tau }_{11}\left({\zeta }_1,{\zeta }_2\right){\zeta }_3+{\tau }_{12}\left({\zeta }_1,{\zeta }_2\right){\zeta }_3^2+{ϝ}_4\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right){\zeta }_4\right],$$

where

$$\begin{array}{ccc}\hfill {\tau }_{10}\left({\zeta }_1,{\zeta }_2\right)& =& -125{\zeta }_1^6+\left(1-|{\zeta }_1{|}^2\right)\left[(1-|{\zeta }_1{|}^2\left)\right(8748{\zeta }_2^2{\zeta }_1^2+8784{\zeta }_2^3|{\zeta }_1{|}^2-7488{\zeta }_2^3+3888{\zeta }_2^4{\overline{{\zeta }_1}}^2)+5400{\zeta }_2^2\overline{{\zeta }_1}{\zeta }_1^3\right.\hfill \\ & & \left.-3300{\zeta }_1^4{\zeta }_2\right],\hfill \\ \hfill {\tau }_{11}\left({\zeta }_1,{\zeta }_2\right)& =& (1-|{\zeta }_2{|}^2)\left(1-|{\zeta }_1{|}^2\right)\left[(-7776\overline{{\zeta }_1}{\zeta }_2^2-1296{\zeta }_2{\zeta }_1)(1-|{\zeta }_1{|}^2)-5400{\zeta }_1^3\right],\hfill \\ \hfill {\tau }_{12}\left({\zeta }_1,{\zeta }_2\right)& =& 3888(1-|{\zeta }_2{|}^2\left)\right(-15-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{{|}^2)}^2,\hfill \\ \hfill {ϝ}_4\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)& =& 62208(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_3{|}^2){\zeta }_2.\hfill \end{array}$$

By putting $\aleph =|{\zeta }_3|,ϵ=|{\zeta }_2|,\zeta =|{\zeta }_1|$ and upon taking $|{\zeta }_4|\le 1$, we get

$$\begin{array}{ccc}\hfill \left|{\mathcal{H}}_{3,1}\left({\kappa }^{-1}\right)\right|& \le & {\displaystyle \frac{1}{8398080}}\left(\left|{\tau }_{10}\left({\zeta }_1,{\zeta }_2\right)\right|+\left|{\tau }_{11}\left({\zeta }_1,{\zeta }_2\right)\right|\aleph +\left|{\tau }_{12}\left({\zeta }_1,{\zeta }_2\right)\right|{\aleph }^2+\left|{ϝ}_4\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)\right|\right)\hfill \\ & \le & {\displaystyle \frac{1}{8398080}}{\Psi }_4\left(\zeta ,ϵ,\aleph \right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\Psi }_4\left(\zeta ,ϵ,\aleph \right)={\varsigma }_{13}\left(\zeta ,ϵ\right)+{\varsigma }_{14}\left(\zeta ,ϵ\right)\aleph +{\varsigma }_{15}\left(\zeta ,ϵ\right){\aleph }^2+{\varsigma }_{16}\left(\zeta ,ϵ\right)(1-{\aleph }^2),\end{array}$$

with

$$\begin{array}{ccc}\hfill {\varsigma }_{13}\left(\zeta ,ϵ\right)& =& \left(1-{\zeta }^2\right)\left\{[3888{ϵ}^4{\zeta }^2+(7488+8784{\zeta }^2){ϵ}^3+8748{ϵ}^2{\zeta }^2](1-{\zeta }^2)+5400{ϵ}^2{\zeta }^4+3300ϵ{\zeta }^4\right\}+125{\zeta }^6,\hfill \\ \hfill {\varsigma }_{14}\left(\zeta ,ϵ\right)& =& \left(1-{ϵ}^2\right)(1-{\zeta }^2)[5400{\zeta }^3+(1-{\zeta }^2)(7776\zeta {ϵ}^2+1296\zeta ϵ)],\hfill \\ \hfill {\varsigma }_{15}\left(\zeta ,ϵ\right)& =& 3888{(1-{\zeta }^2)}^2({ϵ}^2+15)\left(1-{ϵ}^2\right),\hfill \\ \hfill {\varsigma }_{16}\left(\zeta ,ϵ\right)& =& 62208ϵ(1-{ϵ}^2){(1-{\zeta }^2)}^2.\hfill \end{array}$$

Differentiating ${\Psi }_4(\zeta ,ϵ,\aleph )$ with respect to ℵ, we recieve

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\Psi }_4}{\partial \aleph }}& =& {\varsigma }_{14}(\zeta ,ϵ)+2[{\varsigma }_{15}(\zeta ,ϵ)-{\varsigma }_{16}(\zeta ,ϵ)]\aleph \hfill \\ & & ={\varsigma }_{14}(\zeta ,ϵ)+7776\aleph {(1-{\zeta }^2)}^2({ϵ}^2-16ϵ+15)(1-{ϵ}^2)\ge 0.\hfill \end{array}$$

Therefore, ${\Psi }_4(u,v,s)$ attains its maximum value at $\aleph =1$. Thus, we yield

$$\begin{array}{ccc}\hfill {\Psi }_4\left(\zeta ,ϵ,\aleph \right)& \le & {\Psi }_4\left(\zeta ,ϵ,1\right)={\varsigma }_{13}(\zeta ,ϵ)+{\varsigma }_{14}(\zeta ,ϵ)+{\varsigma }_{15}(\zeta ,ϵ)\hfill \\ & =& 58320-11640{\zeta }^2+5400{\zeta }^3+5832{\zeta }^4-5400{\zeta }^5+125{\zeta }^6+(-3888-7776\zeta +11664{\zeta }^2\hfill \\ & +& 15552{\zeta }^3-11664{\zeta }^4-7776{\zeta }^5+3888{\zeta }^6){ϵ}^4+(7488-1296\zeta -6192{\zeta }^2+2592{\zeta }^3-10080{\zeta }^4\hfill \\ & -& 1296{\zeta }^5+8784{\zeta }^6){ϵ}^3+(-54432+7776\zeta +117612{\zeta }^2-20952{\zeta }^3-66528{\zeta }^4+13176{\zeta }^5+3348{\zeta }^6){ϵ}^2\hfill \\ & +& (1296\zeta -2592{\zeta }^3+3300{\zeta }^4+1296{\zeta }^5-3300{\zeta }^6)ϵ={{\rm Y}}_7(\zeta ,ϵ).\hfill \end{array}$$

Consider

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {{\rm Y}}_7}{\partial \zeta }}=-233280\zeta +16200{\zeta }^2+233280{\zeta }^3-27000{\zeta }^4+750{\zeta }^5+(-7776+23328\zeta +46656{\zeta }^2-46656{\zeta }^3\hfill \\ -38880{\zeta }^4+23328{\zeta }^5){ϵ}^4+(-1296-12384\zeta +7776{\zeta }^2-40320{\zeta }^3-6480{\zeta }^4+52704{\zeta }^5){ϵ}^3+(7776+235224\zeta \hfill \\ -62856{\zeta }^2-266112{\zeta }^3+65880{\zeta }^4+20088{\zeta }^5){ϵ}^2+(1296-7776{\zeta }^2+13200{\zeta }^3+6480{\zeta }^4-19800{\zeta }^5)ϵ=0\hfill \\ {\displaystyle \frac{\partial {{\rm Y}}_7}{\partial ϵ}}=-3300{\zeta }^6+1296{\zeta }^5+3300{\zeta }^4-2592{\zeta }^3+1296\zeta +(-15552-31104\zeta +46656{\zeta }^2+62208{\zeta }^3-46656{\zeta }^4\hfill \\ -31104{\zeta }^5+15552{\zeta }^6){ϵ}^3+(22464-3888\zeta -18576{\zeta }^2+7776{\zeta }^3-30240{\zeta }^4-3888{\zeta }^5+26352{\zeta }^6){ϵ}^2+(-108864\hfill \\ +15552\zeta +235224{\zeta }^2-41904{\zeta }^3-133056{\zeta }^4+26352{\zeta }^5+6696{\zeta }^6)ϵ=0\hfill \end{array}\right.$$

By calculating, we have

$\left\{\begin{array}{c}{\zeta }_{1,1}\approx 1,\hfill \\ {ϵ}_{1,1}\approx -1.5227,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,2}\approx -1,\hfill \\ {ϵ}_{1,2}\approx 0.5943,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,3}\approx -1,\hfill \\ {ϵ}_{1,3}\approx -0.8998,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,4}=0,\hfill \\ {ϵ}_{1,4}=0,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,5}\approx 1.0014,\hfill \\ {ϵ}_{1,5}\approx -2.7530,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,6}\approx 1.0207,\hfill \\ {ϵ}_{1,6}\approx -0.6733,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,7}\approx 1.6122,\hfill \\ {ϵ}_{1,7}\approx 1.2088,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,8}\approx 1.9550,\hfill \\ {ϵ}_{1,8}\approx 6.4534,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,9}\approx 5.9181,\hfill \\ {ϵ}_{1,9}\approx 0.2242,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,10}\approx -0.8215,\hfill \\ {ϵ}_{1,10}\approx 1.1057,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,11}\approx -0.9381,\hfill \\ {ϵ}_{1,11}\approx -1.2212,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,12}\approx -0.9767,\hfill \\ {ϵ}_{1,12}\approx -0.2027,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,13}\approx -1.4685,\hfill \\ {ϵ}_{1,13}\approx 1.1426,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,14}\approx -13.5576,\hfill \\ {ϵ}_{1,14}\approx 0.2703.\hfill \end{array}\right.$

Therefore, there is no critical point in $(0,1)\times (0,1).$

(16) For $\zeta =0,$

$$\begin{array}{c}\hfill {{\rm Y}}_7(0,ϵ)=58320-54432{ϵ}^2+7488{ϵ}^3-3888{ϵ}^4\le 58320.\end{array}$$

(17) For $\zeta =1,$

$$\begin{array}{c}\hfill {{\rm Y}}_7(1,ϵ)=125.\end{array}$$

(18) For $ϵ=0,$

$$\begin{array}{c}\hfill {{\rm Y}}_7(\zeta ,0)=58320-116640{\zeta }^2+5400{\zeta }^3+58320{\zeta }^4-5400{\zeta }^5+125{\zeta }^6\le 58320.\end{array}$$

(19) For $ϵ=1,$

$$\begin{array}{c}\hfill {{\rm Y}}_7(\zeta ,1)=12845{\zeta }^6-26652{\zeta }^4+6444{\zeta }^2+7488={\iota }_4\left(\zeta \right)\le {\iota }_4\left(0.3658\right)=7903.8.\end{array}$$

Thus, we have

$$\begin{array}{c}\hfill |{\mathcal{H}}_{3,1}\left({\kappa }^{-1}\right)|\le {\displaystyle \frac{588320}{8398080}}={\displaystyle \frac{1}{144}}.\end{array}$$

Theorem 3.5If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|A_4-A_2{A}_3|\le {\displaystyle \frac{1}{12}}.$$

The result is sharp for the function ${\kappa }_3\left(\xi \right).$

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (41), (42), (43) and Lemma 1.3, we achieve

$$\begin{array}{c}\hfill |A_4-A_2{A}_3|={\displaystyle \frac{1}{12}}|{\beta }_3-{\displaystyle \frac{7}{18}}{\beta }_1{\beta }_2+{\displaystyle \frac{5}{108}}{\beta }_1^3|\le {\displaystyle \frac{1}{12}}.\end{array}$$

Theorem 3.6If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|A_5-A_2{A}_4|\le {\displaystyle \frac{1}{15}}.$$

The result is sharp for the function ${\kappa }_4\left(\xi \right).$

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From Lemma 1.4 and 1.5, (41), (43) and (44), we achieve

$$\begin{array}{cc}\hfill |A_5-A_2{A}_4|& ={\displaystyle \frac{1}{38880}}|-2592{\beta }_4+1404{\beta }_1{\beta }_3+792{\beta }_2^2-642{\beta }_2{\beta }_1^2+200{\beta }_1^4|\hfill \\ \hfill & ={\displaystyle \frac{1}{38880}}|-702(3{\beta }_2{\beta }_1^2-{\beta }_1^4-2{\beta }_1{\beta }_3-{\beta }_2^2+{\beta }_4)-1890{\beta }_4+1464{\beta }_1^2{\beta }_2+72{\beta }_2^2-502{\beta }_1^4|\hfill \\ \hfill & \le {\displaystyle \frac{1}{38880}}[702+1890(1-|{\beta }_1{|}^2-|{\beta }_2{|}^2)+1464(1-|{\beta }_1{|}^2\left)\right|{\beta }_1{|}^2+72|{\beta }_2{|}^2+502|{\beta }_1{|}^4]\hfill \\ \hfill & ={\displaystyle \frac{1}{38880}}[2592-426|{\beta }_1{|}^2-1818|{\beta }_2{|}^2-962|{\beta }_1{|}^4]\le {\displaystyle \frac{2592}{38880}}={\displaystyle \frac{1}{15}}.\hfill \end{array}$$

Theorem 3.7If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|A_5-A_3^2|\le {\displaystyle \frac{1}{15}}.$$

The result is sharp for the function ${\kappa }_4\left(\xi \right).$

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (42), (44), Lemma 1.2, 1.5 and 1.4, we have

$$\begin{array}{cc}\hfill |A_5-A_3^2|& ={\displaystyle \frac{1}{12960}}|864{\beta }_4-648{\beta }_1{\beta }_3-104{\beta }_2^2+244{\beta }_2{\beta }_1^2-75{\beta }_1^4|\hfill \\ \hfill & ={\displaystyle \frac{1}{12960}}|324(3{\beta }_2{\beta }_1^2-{\beta }_2^2-2{\beta }_3{\beta }_1-{\beta }_1^4+{\beta }_4)+540{\beta }_4-249{\beta }_1^2({\beta }_2-{\beta }_1^2)+220{\beta }_2^2-479{\beta }_1^2{\beta }_2|\hfill \\ \hfill & \le {\displaystyle \frac{1}{12960}}[324+540(1-|{\beta }_2{|}^2|-|{\beta }_1{|}^2)+220|{\beta }_2{|}^2+249|{\beta }_1{|}^2+479|{\beta }_1{|}^2|{\beta }_2\left|\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{12960}}[864-291|{\beta }_1{|}^2-320|{\beta }_2{|}^2+479|{\beta }_1{|}^2|{\beta }_2\left|\right].\hfill \end{array}$$

By putting $|{\beta }_1|=x,y=|{\beta }_2|,$ we get

$$\begin{array}{c}\hfill |A_5-A_3^2|\le {\displaystyle \frac{1}{12960}}[864-291x^2-320y^2+479x^{2}y]={\displaystyle \frac{1}{12960}}{\varpi }_2(x,y).\end{array}$$

Consider

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {\varpi }_2}{\partial x}}=-582x+958xy=0,\hfill \\ {\displaystyle \frac{\partial {\varpi }_2}{\partial y}}=-640y+479y^2=0.\hfill \end{array}\right.$$

By calculating, we achieve

$\left\{\begin{array}{c}{x}_1=0,\hfill \\ y_1=0,\hfill \end{array}\right.$$\left\{\begin{array}{c}{x}_2\approx 0.9010,\hfill \\ y_2\approx 0.6075,\hfill \end{array}\right.$$\left\{\begin{array}{c}{x}_3\approx -0.9010,\hfill \\ y_3\approx 0.6075.\hfill \end{array}\right.$

Thus, there is a critical point $(x_2,y_2)$ in $(0,1)\times (0,1).$ So, we have $max\left\{{\varpi }_2(x,y)\right\}={\varpi }_2(0.9010,0.6075)=745.895915.$

(20) For $x=0$

$$\begin{array}{c}\hfill {\varpi }_2(0,y)=864-320y^2\le 864.\end{array}$$

(21) For $y=0,$

$$\begin{array}{c}\hfill {\varpi }_2(x,0)=864-291x^2\le 864\end{array}$$

(22) For $y=1-x^2,$

$$\begin{array}{c}\hfill {\varpi }_2(x,1-x^2)=544+828x^2-799x^4={\iota }_5\left(x\right)\le {\iota }_5\left(0.7198\right)=758.5131.\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill |A_5-A_3^2|\le {\displaystyle \frac{864}{12960}}={\displaystyle \frac{1}{15}}.\end{array}$$

Theorem 3.8If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\delta }_1|\le {\displaystyle \frac{1}{12}},|{\delta }_2|\le {\displaystyle \frac{1}{18}},|{\delta }_3|\le {\displaystyle \frac{1}{24}},\left|{\delta }_4\right|\le {\displaystyle \frac{1}{30}}.$$

These bounds are sharp for the functions ${\kappa }_1\left(\xi \right),{\kappa }_2\left(\xi \right),{\kappa }_3\left(\xi \right)$ and ${\kappa }_4\left(\xi \right)$, respectively.

Proof Let $\kappa \in {\mathcal{R}}_{LE}.$ From (10), (24), (25), (26) and (27), we yield

$$\begin{array}{c}\hfill {\delta }_1=-{\displaystyle \frac{1}{12}}{\beta }_1,\end{array}$$

(47)

$$\begin{array}{c}\hfill {\delta }_2=-{\displaystyle \frac{1}{18}}{\beta }_2+{\displaystyle \frac{1}{144}}{\beta }_1^2,\end{array}$$

(48)

$$\begin{array}{c}\hfill {\delta }_3=-{\displaystyle \frac{1}{24}}{\beta }_3+{\displaystyle \frac{7}{432}}{\beta }_1{\beta }_2-{\displaystyle \frac{7}{2592}}{\beta }_1^3,\end{array}$$

(49)

$$\begin{array}{c}\hfill {\delta }_4=-{\displaystyle \frac{1}{155520}}(5184{\beta }_4-2808{\beta }_1{\beta }_3+1284{\beta }_1^2{\beta }_2-1104{\beta }_2^2-415{\beta }_1^4).\end{array}$$

(50)

From Lemma 1.1, Lemma 1.2, (47) and (48), we have

$$|{\delta }_1|\le {\displaystyle \frac{1}{12}},\left|{\delta }_2\right|\le {\displaystyle \frac{1}{18}}.$$

Rearranging (49), we achieve

$$\begin{array}{c}\hfill |{\delta }_3|={\displaystyle \frac{1}{24}}|{\beta }_3-{\displaystyle \frac{7}{18}}{\beta }_1{\beta }_2+{\displaystyle \frac{7}{108}}{\beta }_1^3|.\end{array}$$

From Lemma 1.3 (with $\nu =-{\displaystyle \frac{7}{18}}$ and $\rho ={\displaystyle \frac{7}{108}}$), we get ${\displaystyle \frac{4}{27}}{\left(\right|\nu |+1)}^3-\left(\right|\nu |+1)=-{\displaystyle \frac{156200}{157464}}<\rho ={\displaystyle \frac{7}{108}}.$ By Lemma 1.3, we obtain

$$|{\delta }_3|\le {\displaystyle \frac{1}{24}}$$

.

From (50), Lemma 1.4 and 1.5, we yield

$$\begin{array}{cc}\hfill |{\delta }_4|& ={\displaystyle \frac{1}{155520}}|1404(3{\beta }_2{\beta }_1^2+{\beta }_4-2{\beta }_3{\beta }_1-{\beta }_1^4-{\beta }_2^2)+3780{\beta }_4-2928{\beta }_2{\beta }_1^2+300{\beta }_2^2+989{\beta }_1^4|\hfill \\ \hfill & \le {\displaystyle \frac{1}{155520}}[1404+3780(1-|{\beta }_1{|}^2-|{\beta }_2{|}^2)+300|{\beta }_2{|}^2+2928|{\beta }_1{|}^2\left|\right(1-|{\beta }_1{|}^2)+989|{\beta }_1{|}^4]\hfill \\ \hfill & ={\displaystyle \frac{1}{155520}}[5184-852|{\beta }_1{|}^2-3480|{\beta }_2{|}^2-1939|{\beta }_1{|}^4]\le {\displaystyle \frac{5184}{155520}}={\displaystyle \frac{1}{30}}.\hfill \end{array}$$

Theorem 3.9If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\delta }_1{\delta }_3-{\delta }_2^2|\le {\displaystyle \frac{1}{324}}.$$

The bound is sharp for the function ${\kappa }_2\left(\xi \right)$.

Proof Let $\kappa \in {\mathcal{R}}_{LE}$. From (47), (48) and (49) we get

$$\begin{array}{c}\hfill |{\delta }_1{\delta }_3-{\delta }_2^2|={\displaystyle \frac{1}{62208}}|-36{\beta }_1^2{\beta }_2-192{\beta }_2^2+216{\beta }_1{\beta }_3+11{\beta }_1^4|.\end{array}$$

(51)

From (51) and Lemma 1.6, we have

$$\begin{array}{ccc}\hfill |{\delta }_1{\delta }_3-{\delta }_2^2|& =& {\displaystyle \frac{1}{62208}}[-192{\zeta }_2^2(1-|{\zeta }_1{|}^2{)}^2-36{\zeta }_1^2(1-|{\zeta }_1{|}^2){\zeta }_2+11{\zeta }_1^4-216{\zeta }_2^2|{\zeta }_1{|}^2(1-|{\zeta }_1{|}^2)\hfill \\ & +& 216{\zeta }_3(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2\left){\zeta }_1\right].\hfill \end{array}$$

By setting $|{\zeta }_2|=ϵ,\zeta =|{\zeta }_1|$, and upon taking $|{\zeta }_3|\le 1$, we obtain

$$\begin{array}{ccc}\hfill |{\delta }_1{\delta }_3-{\delta }_2^2|& \le & {\displaystyle \frac{1}{62208}}[192{(1-{\zeta }^2)}^2{ϵ}^2+36ϵ(1-{\zeta }^2){\zeta }^2+11{\zeta }^4+216{ϵ}^2{\zeta }^2(1-{\zeta }^2)+216\zeta (1-{ϵ}^2)(1-{\zeta }^2)]\hfill \\ & =& {{\rm Y}}_8(\zeta ,ϵ).\hfill \end{array}$$

Differentiating ${{\rm Y}}_8(\zeta ,ϵ)$ with respect to $ϵ$, we yield

$${\displaystyle \frac{\partial {{\rm Y}}_8}{\partial ϵ}}={\displaystyle \frac{1}{62208}}[36(1-{\zeta }^2){\zeta }^2+48({\zeta }^2-9\zeta +8)(1-{\zeta }^2)ϵ]\ge 0.$$

Thus, ${{\rm Y}}_8$ is an increasing function on $ϵ\in [0,1]$. Therefore, we achieve

$$\begin{array}{ccc}\hfill |{\delta }_1{\delta }_3-{\delta }_2^2|& \le & {{\rm Y}}_8(\zeta ,1)={\displaystyle \frac{1}{62208}}(192-132{\zeta }^2-49{\zeta }^4)\le {\displaystyle \frac{192}{62208}}={\displaystyle \frac{1}{324}}.\hfill \end{array}$$

Theorem 3.10If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\delta }_2{\delta }_4-{\delta }_3^2|\le {\displaystyle \frac{1}{576}}.$$

The result is sharp for the function ${\kappa }_4\left(\xi \right).$

Proof Let $\kappa \in {\mathcal{R}}_{LE}$. From (48), (49) and (50), we have

$$\begin{array}{ccc}\hfill {\delta }_2{\delta }_4-{\delta }_3^2& =& {\displaystyle \frac{1}{67184640}}(16488{\beta }_1^2{\beta }_2^2-116640{\beta }_3^2-7932{\beta }_1^4{\beta }_2+755{\beta }_1^6+124416{\beta }_4{\beta }_2+23328{\beta }_1{\beta }_2{\beta }_3-26496{\beta }_2^3\hfill \\ & -& 15552{\beta }_4{\beta }_1^2-6696{\beta }_1^3{\beta }_3)..\hfill \end{array}$$

(52)

From (52) and Lemma 1.6, we achieve

$$\begin{array}{ccc}\hfill {\delta }_2{\delta }_4-{\delta }_3^2& =& {\displaystyle \frac{1}{67184640}}[-15552{\zeta }_3\overline{{\zeta }_1}(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2){\zeta }_2^2+755{\zeta }_1^6+7776{\zeta }_2^4{\overline{{\zeta }_1}}^2(1-|{\zeta }_1{|}^2{)}^2-116640(1-|{\zeta }_2{{|}^2)}^2\hfill \\ & \times & (1-|{\zeta }_1{|}^2{)}^2{\zeta }_3^2-124416{\zeta }_3^2|{\zeta }_2{|}^2(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2+6696{\zeta }_1^3\overline{{\zeta }_1}{\zeta }_2^2(1-|{\zeta }_1{|}^2)+124416{\zeta }_2{\zeta }_4(1-|{\zeta }_1{{|}^2)}^2\hfill \\ & \times & (1-|{\zeta }_3{|}^2\left)\right(1-|{\zeta }_2{|}^2)-6696{\zeta }_3{\zeta }_1^3(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2)-15552{\zeta }_2^3(1-|{\zeta }_1{|}^2){\overline{{\zeta }_1}}^2{\zeta }_1^2+31104{\zeta }_2{\zeta }_3\overline{{\zeta }_1}{\zeta }_1^2\hfill \\ & \times & (1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2)+15552\overline{{\zeta }_2}{\zeta }_1^2(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2){\zeta }_3^2-23328|{\zeta }_1{|}^2{\zeta }_2^3(1-|{\zeta }_1{{|}^2)}^2-15552{\zeta }_1^2{\zeta }_4\hfill \\ & \times & (1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_3{|}^2\left)\right(1-|{\zeta }_1{|}^2)+23328{\zeta }_3{\zeta }_2{\zeta }_1(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2+16488{\zeta }_2^2(1-|{\zeta }_1{{|}^2)}^2{\zeta }_1^2\hfill \\ & -& 26496{\zeta }_2^3(1-|{\zeta }_1{|}^2{)}^3-7932{\zeta }_2{\zeta }_1^4(1-|{\zeta }_1{|}^2)].\hfill \end{array}$$

Therefore, we receive

$${\delta }_2{\delta }_4-{\delta }_3^2={\displaystyle \frac{1}{67184640}}\left[{\tau }_{13}\left({\zeta }_1,{\zeta }_2\right)+{\tau }_{14}\left({\zeta }_1,{\zeta }_2\right){\zeta }_3+{\tau }_{15}\left({\zeta }_1,{\zeta }_2\right){\zeta }_3^2+{ϝ}_5\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right){\zeta }_4\right],$$

where

$$\begin{array}{ccc}\hfill {\tau }_{13}\left({\zeta }_1,{\zeta }_2\right)& =& \left(1-|{\zeta }_1{|}^2\right)\left\{[16488{\zeta }_1^2{\zeta }_2^2-(26496-3168|{\zeta }_1{|}^2){\zeta }_2^3+7776{\overline{{\zeta }_1}}^2{\zeta }_2^4\left]\right(1-|{\zeta }_1{|}^2)-15552{\zeta }_2^3{\zeta }_1^2{\overline{{\zeta }_1}}^2\right.\hfill \\ & & \left.+6696{\zeta }_2^2\overline{{\zeta }_1}{\zeta }_1^3-7932{\zeta }_1^4{\zeta }_2\right\}+755{\zeta }_1^6,\hfill \\ \hfill {\tau }_{14}\left({\zeta }_1,{\zeta }_2\right)& =& (1-|{\zeta }_2{|}^2)\left(1-|{\zeta }_1{|}^2\right)\left[(23328{\zeta }_2{\zeta }_1-15552\overline{{\zeta }_1}{\zeta }_2^2)(1-|{\zeta }_1{|}^2)-6696{\zeta }_1^3+31104{\zeta }_2{\zeta }_1^2\overline{{\zeta }_1}\right],\hfill \\ \hfill {\tau }_{15}\left({\zeta }_1,{\zeta }_2\right)& =& (1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2\left)\right[(1-|{\zeta }_1{|}^2\left)\right(-116640-7776|{\zeta }_2{|}^2)+15552{\zeta }_1^2\overline{{\zeta }_2}],\hfill \\ \hfill {ϝ}_5\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)& =& (1-|{\zeta }_3{|}^2\left)\right(1-|{\zeta }_1{|}^2\left)\right(1-|{\zeta }_2{|}^2\left)\right[124416{\zeta }_2(1-|{\zeta }_1{|}^2)-15552{\zeta }_1^2].\hfill \end{array}$$

By setting $|{\zeta }_3|=\aleph ,|{\zeta }_2|=ϵ,\zeta =|{\zeta }_1|$, and upon taking $|{\zeta }_4|\le 1$, we yield

$$\begin{array}{ccc}\hfill \left|{\delta }_2{\delta }_4-{\delta }_3^2\right|& \le & {\displaystyle \frac{1}{67184640}}\left(\left|{\tau }_{13}\left({\zeta }_1,{\zeta }_2\right)\right|+\left|{\tau }_{14}\left({\zeta }_1,{\zeta }_2\right)\right|\aleph +\left|{\tau }_{15}\left({\zeta }_1,{\zeta }_2\right)\right|{\aleph }^2+\left|{ϝ}_5\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)\right|\right)\hfill \\ & \le & {\displaystyle \frac{1}{67184640}}{\Psi }_5\left(\zeta ,ϵ,\aleph \right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\Psi }_5\left(\zeta ,ϵ,\aleph \right)={\varsigma }_{17}\left(\zeta ,ϵ\right)+{\varsigma }_{18}\left(\zeta ,ϵ\right)\aleph +{\zeta }_{19}\left(\zeta ,ϵ\right){\aleph }^2+{\varsigma }_{20}\left(\zeta ,ϵ\right)(1-{\aleph }^2),\end{array}$$

with

$$\begin{array}{ccc}\hfill {\varsigma }_{17}\left(\zeta ,ϵ\right)& =& \left(1-{\zeta }^2\right)\left\{[16488{\zeta }^2{ϵ}^2+(26469-3168{\zeta }^2){ϵ}^3+7776{\zeta }^2{ϵ}^4](1-{\zeta }^2)+15552{ϵ}^3{\zeta }^4+6696{ϵ}^2{\zeta }^4\right.\hfill \\ & & \left.+7932ϵ{\zeta }^4\right\}+755{\zeta }^6,\hfill \\ \hfill {\varsigma }_{18}\left(\zeta ,\zeta \right)& =& \left(1-{ϵ}^2\right)(1-{\zeta }^2)\left[(23328ϵ\zeta +15552{ϵ}^2\zeta )(1-{\zeta }^2)+31104ϵ{\zeta }^3+6696{\zeta }^3\right],\hfill \\ \hfill {\varsigma }_{19}\left(\zeta ,ϵ\right)& =& \left(1-{\zeta }^2\right)(1-{ϵ}^2)\left[15552ϵ{\zeta }^2+(116640+7776{ϵ}^2)(1-{\zeta }^2)\right],\hfill \\ \hfill {\varsigma }_{20}\left(\zeta ,ϵ\right)& =& (1-{\zeta }^2)(1-{ϵ}^2)\left[124416(1-{\zeta }^2)ϵ+15552{\zeta }^2\right].\hfill \end{array}$$

Note that

$$\begin{array}{ccc}\hfill {\varsigma }_{19}(\zeta ,ϵ))& \le & [(1-{\zeta }^2)(7776{ϵ}^2+116640)+15552{\zeta }^2](1-{ϵ}^2)(1-{\zeta }^2)={\varrho }_2(\zeta ,ϵ),\hfill \end{array}$$

Thus, we receive

$${\Psi }_5\left(\zeta ,ϵ,\aleph \right)\le {\varsigma }_{17}\left(\zeta ,ϵ\right)+{\varsigma }_{18}\left(\zeta ,ϵ\right)s+{\varrho }_2\left(\zeta ,ϵ\right){\aleph }^2+{\varsigma }_{20}\left(\zeta ,ϵ\right)(1-{\aleph }^2)={\Gamma }_2(\zeta ,ϵ,\aleph ).$$

Differentiating ${\Gamma }_2(\zeta ,ϵ,\aleph )$ with respect to ℵ, we yield

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\Gamma }_2}{\partial \aleph }}& =& {\varsigma }_{18}(\zeta ,ϵ)+2\aleph [{\varrho }_2(\zeta ,ϵ)-{\varsigma }_{20}(\zeta ,ϵ)]\hfill \\ & & ={\varsigma }_{18}(\zeta ,ϵ)+15552(1-{ϵ}^2){(1-{\zeta }^2)}^2(15-16ϵ+{ϵ}^2)\aleph \ge 0.\hfill \end{array}$$

Therefore, ${\Gamma }_2(\zeta ,ϵ,\aleph )$ is an increasing function on $\aleph \in [0,1]$. Thus, we have

$$\begin{array}{ccc}\hfill {\Psi }_5\left(\zeta ,ϵ,\aleph \right)& \le & {\Gamma }_2\left(\zeta ,ϵ,\aleph \right)\le {\Gamma }_2\left(\zeta ,ϵ,1\right)={\varsigma }_{17}(\zeta ,ϵ)+{\varsigma }_{18}(\zeta ,ϵ)+{\varrho }_2(\zeta ,ϵ)\hfill \\ & =& 755{\zeta }^6-6696{\zeta }^5+101088{\zeta }^4+6696{\zeta }^3-217728{\zeta }^2+116640+(7776{\zeta }^6-15552{\zeta }^5-23328{\zeta }^4\hfill \\ & +& 31104{\zeta }^3+23328{\zeta }^2-15552\zeta -7776){ϵ}^4+(-18720{\zeta }^6+7776{\zeta }^5+48384{\zeta }^4+15552{\zeta }^3-56160{\zeta }^2\hfill \\ & -& 23328\zeta +26496){ϵ}^3+(9792{\zeta }^6+22248{\zeta }^5-119592{\zeta }^4-37800{\zeta }^3+218664{\zeta }^2+15552\zeta -108864){ϵ}^2\hfill \\ & +& (-7932{\zeta }^6-7776{\zeta }^5+7932{\zeta }^4-15552{\zeta }^3+23328\zeta )ϵ={{\rm Y}}_9(\zeta ,ϵ).\hfill \end{array}$$

The optimal points of ${{\rm Y}}_9(\zeta ,ϵ)$ satisfy the conditions

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {{\rm Y}}_9}{\partial \zeta }}=4530{\zeta }^5-33480{\zeta }^4+404352{\zeta }^3+20088{\zeta }^2-435456\zeta +(46656{\zeta }^5-77760{\zeta }^4-93312{\zeta }^3+93312{\zeta }^2+46656\zeta \hfill \\ -15552){ϵ}^4+(-112320{\zeta }^5+38880{\zeta }^4+193536{\zeta }^3+46656{\zeta }^2-112320\zeta -23328){ϵ}^3+(58752{\zeta }^5+111240{\zeta }^4\hfill \\ -478368{\zeta }^3-113400{\zeta }^2+437328\zeta +15552){ϵ}^2+(-47592{\zeta }^5-38880{\zeta }^4+31728{\zeta }^3-46656{\zeta }^2+23328)ϵ=0\hfill \\ {\displaystyle \frac{\partial {{\rm Y}}_9}{\partial ϵ}}=23328\zeta -15552{\zeta }^3+7932{\zeta }^4-7776{\zeta }^5-7932{\zeta }^6+(31104{\zeta }^6-62208{\zeta }^5-93312{\zeta }^4+124416{\zeta }^3+93312{\zeta }^2\hfill \\ -62208\zeta -31104){ϵ}^3+(-56160{\zeta }^6+23328{\zeta }^5+145152{\zeta }^4+46656{\zeta }^3-168480{\zeta }^2-69984\zeta +79488){ϵ}^2+(19584{\zeta }^6\hfill \\ +44496{\zeta }^5-239184{\zeta }^4-75600{\zeta }^3+437328{\zeta }^2+31104\zeta -217728)ϵ=0\hfill \end{array}\right.$$

By calculating, we get

$\left\{\begin{array}{c}{\zeta }_{1,1}\approx 1,\hfill \\ {ϵ}_{1,1}\approx 1.4152,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,2}\approx 1,\hfill \\ {ϵ}_{1,2}\approx -1.9495,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,3}\approx 1,\hfill \\ {ϵ}_{1,3}\approx -0.4657,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,4}\approx -1,\hfill \\ {ϵ}_{1,4}\approx 0.4562,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,5}\approx -1,\hfill \\ {ϵ}_{1,5}\approx -0.7980,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,6}\approx -1,\hfill \\ {ϵ}_{1,6}\approx 0.3881,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,7}=0,\hfill \\ {ϵ}_{1,7}=0,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,8}\approx 3.4113,\hfill \\ {ϵ}_{1,8}\approx 4.0288,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,9}\approx -0.6885,\hfill \\ {ϵ}_{1,9}\approx 1.1470,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,10}\approx -0.7952,\hfill \\ {ϵ}_{1,10}\approx -8.7920,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,11}\approx -0.9998,\hfill \\ {ϵ}_{1,11}\approx 0.4229,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,12}\approx -1.0332,\hfill \\ {ϵ}_{1,12}\approx -0.4389,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,13}\approx -1.3514,\hfill \\ {ϵ}_{1,13}\approx 3.4059.\hfill \end{array}\right.$

Thus, there is no optimal point in $(0,1)\times (0,1).$

(23) For $\zeta =0$

$$\begin{array}{c}\hfill {{\rm Y}}_9(0,ϵ)=116640-108864{ϵ}^2+26496{ϵ}^3-7776{ϵ}^4\le 116640.\end{array}$$

(24) For $\zeta =1,$

$$\begin{array}{c}\hfill {{\rm Y}}_9(1,ϵ)=755.\end{array}$$

(25) For $ϵ=0,$

$$\begin{array}{c}\hfill {{\rm Y}}_9(\zeta ,0)=116640-217728{\zeta }^2+6696{\zeta }^3+101088{\zeta }^4-6696{\zeta }^5+755{\zeta }^6\le 116640.\end{array}$$

(26) For $ϵ=1,$

$$\begin{array}{c}\hfill {{\rm Y}}_9(\zeta ,1)=-8329{\zeta }^6+14484{\zeta }^4-31896{\zeta }^2+26496\le 26496.\end{array}$$

Thus, we yield

$$\begin{array}{c}\hfill |{\delta }_2{\delta }_4-{\delta }_3^2|\le {\displaystyle \frac{116640}{67184640}}={\displaystyle \frac{1}{576}}.\end{array}$$

Theorem 3.11If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\sigma }_3|\le {\displaystyle \frac{2}{3}},|{\sigma }_4|\le 2,|{\sigma }_5|\le 8.$$

These equalities are sharp for the functions ${\kappa }_2\left(\xi \right),{\kappa }_3\left(\xi \right)$ and ${\kappa }_4\left(\xi \right)$.

Proof Let $\kappa \in {\mathcal{R}}_{LE}$. From (14), (24), (25), (26) and (27), we achieve

$$\begin{array}{c}\hfill {\sigma }_3=-{\displaystyle \frac{2}{3}}{\beta }_2,\end{array}$$

(53)

$$\begin{array}{c}\hfill {\sigma }_4=-2({\beta }_3+{\displaystyle \frac{1}{18}}{\beta }_2{\beta }_1+{\displaystyle \frac{5}{108}}{\beta }_1^3),\end{array}$$

(54)

$$\begin{array}{c}\hfill {\sigma }_5=-{\displaystyle \frac{1}{324}}(2592{\beta }_4+216{\beta }_1{\beta }_3+372{\beta }_1^2{\beta }_2+72{\beta }_2^2-125{\beta }_1^4).\end{array}$$

(55)

From (53), (54), Lemma 1.1 and 1.3, we yield

$$|{\sigma }_3|\le {\displaystyle \frac{2}{3}},\left|{\sigma }_4\right|\le 2.$$

From Lemma 1.4, Lemma 1.5 and (55), we have

$$\begin{array}{cc}\hfill |{\sigma }_5|& ={\displaystyle \frac{1}{324}}|108({\beta }_4+{\beta }_2^2+2{\beta }_3{\beta }_1+{\beta }_1^4+3{\beta }_2{\beta }_1^2)+2484{\beta }_4-36{\beta }_2^2+48{\beta }_2{\beta }_1^2-233{\beta }_1^4|\hfill \\ \hfill & \le {\displaystyle \frac{1}{324}}[108+2484(1-|{\beta }_1{|}^2-|{\beta }_2{|}^2)+36|{\beta }_2{|}^2+48(1-|{\beta }_1{|}^2\left)\right|{\beta }_1{|}^2+233|{\beta }_1{|}^4]\hfill \\ \hfill & ={\displaystyle \frac{1}{324}}[2592-2436|{\beta }_1{|}^2-2448|{\beta }_2{|}^2+185|{\beta }_1{|}^4].\hfill \end{array}$$

By setting $x=|{\beta }_1|,|{\beta }_2|=y,$ we have

$$\begin{array}{c}\hfill |{\sigma }_5|\le {\displaystyle \frac{1}{324}}[2592-2436x^2-2488y^2+185x^4]={\displaystyle \frac{1}{324}}{\varpi }_3(x,y).\end{array}$$

Consider

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {\varpi }_3}{\partial x}}=-4872x+740x^3=0,\hfill \\ {\displaystyle \frac{\partial {\varpi }_3}{\partial y}}=-4896y=0.\hfill \end{array}\right.$$

By calculating, we achieve

$\left\{\begin{array}{c}{x}_1=0,\hfill \\ y_1=0,\hfill \end{array}\right.$$\left\{\begin{array}{c}{x}_2\approx 0.9010,\hfill \\ y_2\approx 0.6075,\hfill \end{array}\right.$$\left\{\begin{array}{c}{x}_3\approx -0.9010,\hfill \\ y_3\approx 0.6075\hfill \end{array}\right.$

Thus, there is a optimal point $(x_2,y_2)$ in $(0,1)\times (0,1).$ So, we have $max\left\{{\varpi }_3(x,y)\right\}={\varpi }_3(0.9010,0.6075)=745.895915.$

(27) For $x=0$

$$\begin{array}{c}\hfill {\varpi }_3(0,y)=2592-2488y^2\le 2592.\end{array}$$

(28) For $y=0,$

$$\begin{array}{c}\hfill {\varpi }_3(x,0)=2592-2436x^2+185x^4\le 2592.\end{array}$$

(29) For $y=1-x^2,$

$$\begin{array}{c}\hfill {\varpi }_3(x,1-x^2)=144+2460x^2-2263x^4={\iota }_6\left(x\right)\le {\iota }_6\left(0.7372\right)=812.5373.\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill |{\sigma }_5|\le {\displaystyle \frac{2592}{324}}=8.\end{array}$$

Theorem 3.12If $\kappa \in {\mathcal{R}}_{LE}$, then

$$|{\sigma }_3{\sigma }_5-{\sigma }_4^2|\le 4.$$

The equality is sharp for the functions ${\kappa }_3\left(\xi \right)$.

Proof Let $\kappa \in {\mathcal{R}}_{LE}$. From (53), (54) and (55), we achieve

$$\begin{array}{c}\hfill {\sigma }_3{\sigma }_5-{\sigma }_4^2={\displaystyle \frac{1}{2916}}[-810{\beta }_1^4{\beta }_2-11664{\beta }_3^2+2196{\beta }_1^2{\beta }_2^2-25{\beta }_1^6+15552{\beta }_4{\beta }_2-1080{\beta }_3{\beta }_1^3+432{\beta }_2^3].\end{array}$$

(56)

From Lemma 1.6 and (56), we have

$$\begin{array}{ccc}\hfill {\sigma }_3{\sigma }_5-{\sigma }_4^2& =& {\displaystyle \frac{1}{2916}}\left[432\right(1-|{\zeta }_1{|}^2{)}^3{\zeta }_2^3-810{\zeta }_1^4{\zeta }_2(1-|{\zeta }_1{|}^2)+2196{\zeta }_1^2{(1-{\zeta }_1{|}^2)}^2{\zeta }_2^2+3888{\zeta }_2^4{\overline{{\zeta }_1}}^2(1-|{\zeta }_1{{|}^2)}^2\hfill \\ & -& 15552{\zeta }_3^2(1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_2{|}^2\left)\right|{\zeta }_2{|}^2-25{\zeta }_1^6-7776\overline{{\zeta }_1}{\zeta }_2^2(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{|}^2{)}^2{\zeta }_3+15552{\zeta }_2{\zeta }_4(1-|{\zeta }_2{|}^2)\hfill \\ & \times & (1-|{\zeta }_1{|}^2{)}^2(1-|{\zeta }_3{|}^2)-11664(1-|{\zeta }_2{|}^2{)}^2(1-|{\zeta }_1{|}^2{)}^2{\zeta }_3^2+1080(1-|{\zeta }_1{|}^2){\zeta }_2^2{\zeta }_1^3\overline{{\zeta }_1}-1080{\zeta }_1^3{\zeta }_3(1-|{\zeta }_1{|}^2)\hfill \\ & \times & (1-|{\zeta }_2{|}^2)-25{\zeta }_1^6].\hfill \end{array}$$

Thus, we yield

$${\sigma }_3{\sigma }_5-{\sigma }_4^2={\displaystyle \frac{1}{2916}}\left[{\tau }_{16}\left({\zeta }_1,{\zeta }_2\right)+{\tau }_{17}\left({\zeta }_1,{\zeta }_2\right){\zeta }_3+{\tau }_{18}\left({\zeta }_1,{\zeta }_2\right){\zeta }_3^2+{ϝ}_6\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right){\zeta }_4\right],$$

where

$$\begin{array}{ccc}\hfill {\tau }_{16}\left({\zeta }_1,{\zeta }_2\right)& =& \left(1-|{\zeta }_1{|}^2\right)\left\{[2916{\zeta }_1^2{\zeta }_2^2+432(1-|{\zeta }_1{|}^2){\zeta }_2^3+3888{\zeta }_2^4{\overline{{\zeta }_1}}^2\left]\right(1-|{\zeta }_1{|}^2)+1080{\zeta }_1^3\overline{{\zeta }_1}{\zeta }_2^2-810{\zeta }_2{\zeta }_1^4\right\}-25{\zeta }_1^6,\hfill \\ \hfill {\tau }_{17}\left({\zeta }_1,{\zeta }_2\right)& =& 216(1-|{\zeta }_1{|}^2)\left(1-|{\zeta }_2{|}^2\right)\left[-36(1-|{\zeta }_1{|}^2)\overline{{\zeta }_1}{\zeta }_2^2-5{\zeta }_1^3\right],\hfill \\ \hfill {\tau }_{18}\left({\zeta }_1,{\zeta }_2\right)& =& -3888(1-|{\zeta }_2{|}^2\left)\right(3+|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{{|}^2)}^2,\hfill \\ \hfill {ϝ}_6\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)& =& 15552(1-|{\zeta }_3{|}^2\left)\right(1-|{\zeta }_2{|}^2\left)\right(1-|{\zeta }_1{{|}^2)}^2{\zeta }_2.\hfill \end{array}$$

By setting $ϵ=|{\zeta }_2|,\zeta =|{\zeta }_1|,\aleph =|{\zeta }_3|$ and upon taking $|{\zeta }_4|\le 1$, we yield

$$\begin{array}{ccc}\hfill {\sigma }_3{\sigma }_5-{\sigma }_4^2& \le & {\displaystyle \frac{1}{2361960}}\left(\left|{\tau }_{16}\left({\zeta }_1,{\zeta }_2\right)\right|+\left|{\tau }_{17}\left({\zeta }_1,{\zeta }_2\right)\right|\aleph +\left|{\tau }_{18}\left({\zeta }_1,{\zeta }_2\right)\right|{\aleph }^2+\left|{ϝ}_6\left({\zeta }_1,{\zeta }_2,{\zeta }_3\right)\right|\right)\hfill \\ & \le & {\displaystyle \frac{1}{2916}}{\Psi }_6\left(\zeta ,ϵ,\aleph \right),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\Psi }_6\left(\zeta ,ϵ,\aleph \right)={\varsigma }_{21}\left(\zeta ,ϵ\right)+{\varsigma }_{22}\left(\zeta ,ϵ\right)\aleph +{\varsigma }_{23}\left(\zeta ,ϵ\right){\aleph }^2+{\varsigma }_{24}\left(\zeta ,ϵ\right)(1-{\aleph }^2),\end{array}$$

with

$$\begin{array}{ccc}\hfill {\varsigma }_{21}\left(\zeta ,ϵ\right)& =& 25{\zeta }^6+\left(1-{\zeta }^2\right)\left\{\left[3888{ϵ}^4{\zeta }^2+432(1-{\zeta }^2){ϵ}^3+2916{\zeta }^2{ϵ}^2\right](1-{\zeta }^2)+1080{\zeta }^4{ϵ}^2+810{\zeta }^4ϵ\right\},\hfill \\ \hfill {\varsigma }_{22}\left(\zeta ,ϵ\right)& =& 216(1-{\zeta }^2)\left(1-{ϵ}^2\right)\left[36(1-{\zeta }^2)\zeta {ϵ}^2+5{\zeta }^3\right],\hfill \\ \hfill {\varsigma }_{23}\left(\zeta ,ϵ\right)& =& 3888({ϵ}^2+3)(1-{ϵ}^2){\left(1-{\zeta }^2\right)}^2,\hfill \\ \hfill {\varsigma }_{24}\left(\zeta ,ϵ\right)& =& 15552(ϵ-{ϵ}^3){\left(1-{\zeta }^2\right)}^2.\hfill \end{array}$$

Differentiating ${\Psi }_6(\zeta ,ϵ,\aleph )$ with respect to ℵ, we yield

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\Psi }_6}{\partial \aleph }}& =& {\varsigma }_{22}(\zeta ,ϵ)+2[{\varsigma }_{23}(\zeta ,ϵ)-{\varsigma }_{24}(\zeta ,ϵ)]\aleph \hfill \\ & & ={\varsigma }_{22}(\zeta ,ϵ)+7776\aleph (3-4ϵ+{ϵ}^2)(1-{ϵ}^2){(1-{\zeta }^2)}^2\ge 0.\hfill \end{array}$$

Thus, ${\Psi }_6(\zeta ,ϵ,\aleph )$ attains its maximum value at $\aleph =1$. Therefore, we achieve

$$\begin{array}{ccc}\hfill {\Psi }_6\left(\zeta ,ϵ,\aleph \right)& \le & {\Psi }_6\left(\zeta ,ϵ,1\right)={\varsigma }_{21}(\zeta ,ϵ)+{\varsigma }_{22}(\zeta ,ϵ)+{\varsigma }_{23}(\zeta ,ϵ)\hfill \\ & =& 25{\zeta }^6-1080{\zeta }^5+11664{\zeta }^4+1080{\zeta }^3-23328{\zeta }^2+11664+(3888{\zeta }^6-7776{\zeta }^5-11664{\zeta }^4+15552{\zeta }^3\hfill \\ & +& 11664{\zeta }^2-7776\zeta -3888){ϵ}^4+(-432{\zeta }^6+1296{\zeta }^4-1296{\zeta }^2+432){ϵ}^3+(1836{\zeta }^6+8856{\zeta }^5-12528{\zeta }^4\hfill \\ & -& 16632{\zeta }^3+18468{\zeta }^2+7776\zeta -7776){ϵ}^2+(-810{\zeta }^6+810{\zeta }^4)ϵ={{\rm Y}}_{10}(\zeta ,ϵ).\hfill \end{array}$$

The optimal points of ${{\rm Y}}_{10}(\zeta ,ϵ)$ satisfy the conditions

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {{\rm Y}}_{10}}{\partial \zeta }}=150{\zeta }^5-5400{\zeta }^4+46656{\zeta }^3+3240{\zeta }^2-46656\zeta +(23328{\zeta }^5-38880{\zeta }^4-46656{\zeta }^3+46656{\zeta }^2+23328\zeta \hfill \\ -7776){ϵ}^4+(-2592{\zeta }^5+5184{\zeta }^3-2592\zeta ){ϵ}^3+(11016{\zeta }^5+44280{\zeta }^4-50112{\zeta }^3-49896{\zeta }^2+36936\zeta +7776){ϵ}^2\hfill \\ +(-4860{\zeta }^5+3240{\zeta }^3)ϵ=0\hfill \\ {\displaystyle \frac{\partial {{\rm Y}}_{10}}{\partial ϵ}}=810{\zeta }^4-810{\zeta }^6+(15552{\zeta }^6-31104{\zeta }^5-46656{\zeta }^4+62208{\zeta }^3+46656{\zeta }^2-31104\zeta -15552){ϵ}^3+(-1296{\zeta }^6\hfill \\ +3888{\zeta }^4-3888{\zeta }^2+1296){ϵ}^2+(3672{\zeta }^6+17712{\zeta }^5-25056{\zeta }^4-33264{\zeta }^3+36936{\zeta }^2+15552\zeta -15552)ϵ=0\hfill \end{array}\right.$$

By calculating, we yield

$\left\{\begin{array}{c}{\zeta }_{1,1}\approx 1,\hfill \\ {ϵ}_{1,1}\approx -1.2407,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,2}\approx -1,\hfill \\ {ϵ}_{1,2}\approx 0.5674,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,3}\approx -1,\hfill \\ {ϵ}_{1,3}\approx -0.9424,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,4}=0,\hfill \\ {ϵ}_{1,4}=0,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,5}\approx 1.0012,\hfill \\ {ϵ}_{1,5}\approx -2.2626,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,6}\approx 1.0101,\hfill \\ {ϵ}_{1,6}\approx -1.1499,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,7}\approx 6.8109,\hfill \\ {ϵ}_{1,7}\approx -0.1397,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,8}\approx -0.7316,\hfill \\ {ϵ}_{1,8}\approx 1.1217,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,9}\approx -0.7527,\hfill \\ {ϵ}_{1,9}\approx -1.1329,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,10}\approx -0.9756,\hfill \\ {ϵ}_{1,10}\approx -0.2579,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,11}\approx -1.0805,\hfill \\ {ϵ}_{1,11}\approx 1.3360,\hfill \end{array}\right.$$\left\{\begin{array}{c}{\zeta }_{1,12}\approx -16.6313,\hfill \\ {ϵ}_{1,12}\approx 0.2474.\hfill \end{array}\right.$

Thus, there is no optimal point in $(0,1)\times (0,1).$

(30) For $\zeta =0$

$$\begin{array}{c}\hfill {{\rm Y}}_{10}(0,ϵ)=11664-7776{ϵ}^2+432{ϵ}^3-3888{ϵ}^4\le 11664.\end{array}$$

(31) For $\zeta =1,$

$$\begin{array}{c}\hfill {{\rm Y}}_{10}(1,ϵ)=25.\end{array}$$

(32) For $ϵ=0,$

$$\begin{array}{c}\hfill {{\rm Y}}_{10}(\zeta ,0)=11664-23328{\zeta }^2+1080{\zeta }^3+11664{\zeta }^4-1080{\zeta }^5+25{\zeta }^6\le 11664.\end{array}$$

(33) For $ϵ=1,$

$$\begin{array}{c}\hfill {{\rm Y}}_{10}(\zeta ,1)=4507{\zeta }^6-10422{\zeta }^4+5508{\zeta }^2+432={\iota }_7\left(\zeta \right)\le {\iota }_7\left(0.5819\right)=1277.1.\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill |{\sigma }_3{\sigma }_5-{\sigma }_4^2|\le {\displaystyle \frac{11664}{2916}}=4.\end{array}$$