Strong Convexity Based Improvements of Jensen-Mercer Inequalities in Functional, Integral and Probabilistic Settings

1. Introduction

The notion of convexity plays an important role in many areas of mathematics, especially in analysis, optimization and inequality theory. Convex functions possess a number of useful properties which make them suitable for studying various analytical and functional problems. Because of their simple geometric interpretation and wide applicability, convexity methods are frequently used in the derivation of different inequalities and estimates.

A function $\Omega :\mathbb{I}\subseteq \mathbb{R}\to \mathbb{R}$ is said to be convex if

$$\Omega (\lambda \xi +(1-\lambda \left)\nu \right)⩽\lambda \Omega \left(\xi \right)+(1-\lambda )\Omega \left(\nu \right)$$

(1)

for all $\xi ,\nu \in \mathbb{I}$ and $\lambda \in [0,1]$.

Inequalities related to convex functions have been studied extensively and have found applications in mathematical analysis, probability theory and optimization. Among the most important results in this area, an essential tool in the study of convex functions is the Jensen inequality

$$\Omega \left(\overline{\xi }\right)\le {\displaystyle \sum _{i=1}^{\kappa }}{a}_i\Omega \left({\xi }_i\right)$$

(2)

which holds for every convex function $\Omega :\mathbb{I}\subseteq \mathbb{R}\to \mathbb{R}$, every vector $\xi =\left({\xi }_1,...,{\xi }_{\kappa }\right)\in {\mathbb{I}}^{\kappa }$ and every nonnegative $\kappa$-tuple $\mathbf{a}=(a_1,...,a_{\kappa })$ satisfying ${\sum }_{i=1}^{\kappa }{a}_i=1,$ where $\overline{\xi }={\sum }_{i=1}^{\kappa }{a}_i{\xi }_i.$

Jensen’s inequality is important because it allows us to compare the value of a convex function at an average point with the average of the values of that function. This is why it has many applications in mathematical evaluations and proofs. Since it is very flexible and can be used in various areas of mathematics, numerous upgrades and extensions have been developed. Today, Jensen’s inequalities play an important role in inequality theory, especially in the study of means, optimization and variational methods.

The geometric structure of convex functions also leads naturally to various reflected and generalized forms of Jensen’s inequality. Among them, particular attention has been devoted to the Jensen-Mercer inequality, introduced by A. McD. Mercer in [1]. This inequality may be viewed as a reflected counterpart of the classical Jensen inequality and is based on the transformation $x\mapsto m+n-x$.

More precisely, for a convex function $\Omega :\mathbb{I}\to \mathbb{R}$, numbers $m,n\in \mathbb{I}$ with $m<n,$ a vector $\xi =\left({\xi }_1,...,{\xi }_{\kappa }\right)\in {[m,n]}^{\kappa }$ and a nonnegative $\kappa$-tuple $\mathbf{a}=(a_1,...,a_{\kappa })$ such that ${\sum }_{i=1}^{\kappa }{a}_i=1$ and $\overline{\xi }={\sum }_{i=1}^{\kappa }{a}_i{\xi }_i,$ the Jensen-Mercer inequality states that

$$\Omega \left(m+n-\overline{\xi }\right)\le \Omega \left(m\right)+\Omega \left(n\right)-{\displaystyle \sum _{i=1}^{\kappa }}{a}_i\Omega \left({\xi }_i\right).$$

(3)

The reflected form appearing in Jensen-Mercer inequalities allows a finer control of endpoint contributions than the classical Jensen framework. This becomes particularly effective in the strongly convex setting, a stronger form of convexity.

A function $\Omega :\mathbb{I}\subseteq \mathbb{R}\to \mathbb{R}$ is called strongly convex with modulus $c>0,$ if

$$\Omega (\lambda \xi +(1-\lambda )\nu )⩽\lambda \Omega \left(\xi \right)+(1-\lambda )\Omega \left(\nu \right)-c\lambda (1-\lambda ){(\xi -\nu )}^2$$

(4)

for all $\xi ,\nu \in \mathbb{I}$ and $\lambda \in [0,1].$

Strong convexity may be viewed as a natural strengthening of ordinary convexity obtained by adding a quadratic correction term $c\lambda (1-\lambda ){(\xi -\nu )}^2$ to the defining inequality. This additional term allows better control of the function and often leads to stronger versions of classical inequalities. For this reason, many results involving convex functions have been extended to the strongly convex setting. Strong convexity concept provides sharper estimates in various optimization problems. Together with ordinary convex functions, they are being increasingly studied and applied in the recent literature because of their useful analytical properties.

Throughout the paper, the notation $\Omega \in S{C}_c\left(\mathbb{I}\right)$ means that $\Omega :\mathbb{I}\to \mathbb{R}$ is strongly convex on $\mathbb{I}$ with modulus $c>0$.

Every strongly convex function is necessarily convex, while the converse statement is generally false. Several useful characterizations of strong convexity are summarized in the following lemmas (see [5,15,16]).

Lemma 1.

The function $\Omega \in S{C}_c\left(\mathbb{I}\right)$ if and only if the mapping $h:\mathbb{I}\to \mathbb{R},$ defined by $h\left(t\right)=\Omega \left(t\right)-c{t}^2,$ is convex.

Lemma 2.

Suppose that $\Omega :\mathbb{I}\to \mathbb{R}$ is twice differentiable. Then $\Omega \in S{C}_c\left(\mathbb{I}\right)$ if and only if ${\Omega }^{\prime \prime }\left(t\right)⩾2c$ for every $t\in \mathbb{I}$.

In Section 3, we focus in particular on the following example of strongly convex functions.

Example 1.

Let $\Omega \left(t\right)=t^q$, where $q\in \mathbb{R}$ and $t\in [m,n]\subset (0,\infty )$. Since

$${\Omega }^{\prime \prime }\left(t\right)=q(q-1)t^{q-2},$$

Lemma 2, implies that the modulus of strong convexity is determined by

$${\displaystyle \frac{1}{2}}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2}$$

with the explicit representation

$$\underset{t\in [m,n]}{min}{t}^{q-2}=\left\{\begin{array}{cc}{m}^{q-2},\hfill & q\ge 2,\hfill \\ n^{q-2},\hfill & q<2.\hfill \end{array}\right.$$

Consequently, one may choose

$$c={\displaystyle \frac{1}{2}}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2},$$

that is,

$$c=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}q(q-1)m^{q-2},\hfill & q\ge 2,\hfill \\ {\displaystyle \frac{1}{2}}q(q-1)n^{q-2},\hfill & q<2.\hfill \end{array}\right.$$

Strongly convex functions satisfy the strengthened Jensen inequality

$$\Omega \left(\overline{\xi }\right)\le {\displaystyle \sum _{i=1}^{\kappa }}{a}_i\Omega \left({\xi }_i\right)-c{\displaystyle \sum _{i=1}^{\kappa }}{a}_i{({\xi }_i-\overline{\xi })}^2.$$

(5)

For $c=0$, inequality (5) becomes the classical Jensen inequality (2). This means that strong convexity can be seen as a stronger version of ordinary convexity because of the additional quadratic term. Moreover, for $\kappa =2$, the difference between the two sides of (5) can be written as

$${\Delta }_{\Omega }(c;\xi ,\nu )=\Omega \left(\xi \right)+\Omega \left(\nu \right)-2\Omega \left({\displaystyle \frac{\xi +\nu }{2}}\right)-{\displaystyle \frac{c}{2}}{\left(\nu -\xi \right)}^2.$$

(6)

In the special case $c=0$, we obtain

$${\Delta }_{\Omega }(\xi ,\nu )=\Omega \left(\xi \right)+\Omega \left(\nu \right)-2\Omega \left({\displaystyle \frac{\xi +\nu }{2}}\right).$$

(7)

The quantities ${\Delta }_{\Omega }(c;\xi ,\nu )$ and ${\Delta }_{\Omega }(\xi ,\nu )$ measure the deviation from linear behavior at the midpoint and naturally arise in refined forms of Jensen type inequalities. Their nonnegativity follows directly from Jensen’s inequality for strongly convex and convex functions, respectively.

Both Jensen’s inequality (2) and the Jensen–Mercer inequality (3) can be extended to positive linear functionals, which gives a useful framework for deriving different functional inequalities. Let L be a linear class of real-valued functions defined on a nonempty set T, containing the unit function $1\in L,$ where $1\left(t\right)=1$ for every $t\in T$. We consider a positive normalized linear functional $F:L\to \mathbb{R}$ (briefly, $F\in \mathcal{L}(L,\mathbb{R})$) characterized by the following properties:

• $F(\alpha g+\beta h)=\alpha F\left(g\right)+\beta F\left(h\right),$ for all $g,h\in L$ and $\alpha ,\beta \in \mathbb{R},$
• if $h\in L$ and $h\left(t\right)⩾0$ for all $t\in T,$ then $F\left(h\right)⩾0$, together with the normalization condition $F\left(1\right)=1.$

This functional approach connects discrete, integral and probabilistic versions of Jensen type inequalities within one general framework. In particular, if $F\in \mathcal{L}(L,\mathbb{R})$, $\Omega$ is continuous and convex on $[m,n]$, and $h\in L$ with $\Omega \left(h\right)\in L$, then Jessen’s inequality has the form

$$\Omega \left(F\right(h\left)\right)⩽F\left(\Omega \right(h\left)\right).$$

(8)

Conversely, the associated converse Jessen inequality provides the estimate

$$F\left(\Omega \left(h\right)\right)⩽{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right).$$

(9)

Hence, inequalities (8) and (9) together yield two-sided bounds for the quantity $F\left(\Omega \right(h\left)\right)-\Omega \left(F\right(h\left)\right),$ which measures the deviation from equality in Jensen’s inequality.

The corresponding functional Jensen–Mercer inequality is given by

$$\Omega \left(m+n-F\left(h\right)\right)\le \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right),$$

(10)

and is based on the reflected transformation $h\mapsto m+n-h.$ This reflected form gives estimates that complement those obtained from the classical Jensen inequality.

For the purposes of several results below, we additionally assume that L is a lattice, meaning that:

• if $h,g\in L,$ then $min\{h,g\}\in L$ and $max\{h,g\}\in L.$

We will explicitly emphasize when L is assumed to be a lattice, whenever this property is required.

Remark 1.

The ordered space $\left({\mathbb{R}}^T,⩽\right),$ equipped with the pointwise order, forms a lattice. Moreover, a subspace $X\subseteq {\mathbb{R}}^T$ is a lattice if and only if $t\in X$ implies $\left|t\right|\in X$. Indeed, for every $t\in X,$ the positive and negative parts $t_{-}$ and $t_{+}$ are determined through lattice operations, where $\left|t\right|\left(s\right)=\left|t\right(s\left)\right|,$ $t_{-}=-min\{0,t\left(s\right)\},$$t_{+}=max\{0,t\left(s\right)\}$ for $s\in T,$ while $t_{+}+t_{-}=\left|t\right|,$$t_{+}-t_{-}=t.$ In addition, the lattice operations can be represented by

$$min\{t,u\}={\displaystyle \frac{1}{2}}(t+u-|t-u\left|\right),max\{t,u\}={\displaystyle \frac{1}{2}}(t+u+|t-u\left|\right).$$

In recent years, numerous strengthened and refined forms of the Jensen-Mercer inequality have been investigated, especially in the setting of positive linear functionals. Special attention has been given to improvements of Jessen’s inequality and its converse obtained through Jensen-Mercer type methods. One important improvement was proved in [2], and is stated below,

$$\begin{array}{cc}\hfill \Omega \left(m+n-F\left(h\right)\right)& ⩽F\left(\Omega (m+n-h)\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \right(h\left)\right),\hfill \end{array}$$

(11)

and holds for all continuous convex function $\Omega :[m,n]\to \mathbb{R},$$F\in \mathcal{L}(L,\mathbb{R})$ and $h\in L$ such that $\Omega \left(h\right),\Omega (m+n-h)\in L.$

A key example of improvement is another sharp Jessen-Mercer type inequalities [3],

$$\begin{array}{cc}\hfill & \Omega (m+n-F(h\left)\right)\hfill \\ \hfill & \le {\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)-\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)\hfill \\ \hfill & -\left[1-{\displaystyle \frac{1}{n-m}}\left(F\left(\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)-\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right)\right]{\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-\left[1-{\displaystyle \frac{2}{n-m}}F\left(\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)\right]{\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right),\hfill \end{array}$$

(12)

with

$${\Delta }_{\Omega }(m,n)=\Omega \left(m\right)+\Omega \left(n\right)-2\Omega \left({\displaystyle \frac{m+n}{2}}\right)\ge 0,$$

(13)

valid for all continuous convex function $\Omega :[m,n]\to \mathbb{R},$$F\in \mathcal{L}(L,\mathbb{R})$ on lattice L and $h\in L$ such that $\Omega \left(h\right),\Omega (m+n-h)\in L.$

Both inequalities (11) and (12) provide the intermediate bounds for the Jessen-Mercer inequality (10).

The classical Jessen (8) and the converse Jessen inequality (9) can be further sharpened by virtue of the stronger properties of strongly convex functions. The first refinement results were obtained in the paper [4]. Subsequently, exploiting the stronger structure of strongly convex functions, further refinements of Jessen’s and converse Jessen’s inequalities were established more recent in [5], which we cite and present here.

Let $\mathbb{I}\subseteq \mathbb{R}$ be an open interval, $\Omega \in S{C}_c\left(\mathbb{I}\right)$ and $0⩽K⩽1.$ Further, let $F\in \mathcal{L}(L,\mathbb{R}),$$h\in L,$ such that $h^2,\Omega \left(h\right)\in L.$ Then the following refinement holds:

$$\begin{array}{cc}\hfill \Omega \left(F\left(h\right)\right)& ⩽\Omega \left(F\left(h\right)\right)+K\left[F\left(\Omega \left(h\right)\right)-\Omega \left(F\left(h\right)\right)-c\left(F\left(h^2\right)-{\left(F\left(h\right)\right)}^2\right)\right]\hfill \\ \hfill & ⩽F\left(\Omega \left(h\right)\right)-c\left(F\left(h^2\right)-{\left(F\left(h\right)\right)}^2\right)\hfill \\ \hfill & ⩽F\left(\Omega \left(h\right)\right)\hfill \end{array}$$

(14)

In the same paper [5], the improvement of the converse Jessen inequality (9) was proved. Let $\mathbb{I}\subseteq \mathbb{R}$ be an open interval, $m,n\in \mathbb{I}$ with $m<n$ and $\Omega \in S{C}_c\left(\mathbb{I}\right).$ Assume that $F\in \mathcal{L}(L,\mathbb{R})$ on a lattice $L,$$h\in L$ with values in $[m,n]$ such that $h^2,\Omega \left(h\right)\in L.$ Then

$$\begin{array}{cc}\hfill F\left(\Omega \right(h\left)\right)& ⩽{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)-c{R}_{F\left(h\right)}(m,n)\hfill \\ \hfill & -F\left(h_{min}\right){\Delta }_{\Omega }(c;m,n),\hfill \end{array}$$

(15)

where $h_{min}$ is defined by

$$h_{min}=min\left\{{\displaystyle \frac{n-h\left(x\right)}{n-m}},{\displaystyle \frac{h\left(x\right)-m}{n-m}}\right\}={\displaystyle \frac{1}{2}}1-{\displaystyle \frac{1}{n-m}}\left|h-{\displaystyle \frac{m+n}{2}}1\right|\ge 0,$$

(16)

$$F\left(h_{min}\right)={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}F\left(\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)\ge 0,$$

(17)

$$R_{F\left(h\right)}(m,n)={\displaystyle \frac{F\left(h\right)-m}{n-m}}{n}^2+{\displaystyle \frac{n-F\left(h\right)}{n-m}}{m}^2-F\left(h^2\right)\ge 0$$

(18)

and

$${\Delta }_{\Omega }(c;m,n)=\Omega \left(m\right)+\Omega \left(n\right)-2\Omega \left({\displaystyle \frac{m+n}{2}}\right)-{\displaystyle \frac{c}{2}}{\left(m-n\right)}^2\ge 0.$$

(19)

Specially for $c=0,$ inequality (15) reduces to

$$F\left(\Omega \left(h\right)\right)⩽{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)-F\left(h_{min}\right){\Delta }_{\Omega }(m,n),$$

(20)

where $F\left(h_{min}\right)$ and ${\Delta }_{\Omega }(m,n)$ are given by (17), (16), respectively.

Remark 2.

Note that it holds

$$\begin{array}{cc}\hfill R_{F\left(h\right)}(m,n)& ={\displaystyle \frac{F\left(h\right)-m}{n-m}}{n}^2+{\displaystyle \frac{n-F\left(h\right)}{n-m}}{m}^2-F\left(h^2\right)\hfill \\ \hfill & ={\displaystyle \frac{F\left(h\right)n^2-m{n}^2+n{m}^2-F\left(h\right)m^2-F\left(h^2\right)n-F\left(h^2\right)m}{n-m}}\hfill \\ \hfill & ={\displaystyle \frac{F\left(h\right)(n^2-m^2)-mn(n-m)-F\left(h^2\right)(n-m)}{n-m}}\hfill \\ \hfill & =F\left(h\right)(n+m)-mn-F\left(h^2\right)\hfill \\ \hfill & =(n-F\left(h\right))(F\left(h\right)-m)+{\left(F\left(h\right)\right)}^2-F\left(h^2\right).\hfill \end{array}$$

That enables as to write (15) in the equivalent forms

$$\begin{array}{cc}\hfill F\left(\Omega \right(h\left)\right)& ⩽{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & -c\left[(n-F\left(h\right))(F\left(h\right)-m)+{\left(F\left(h\right)\right)}^2-F\left(h^2\right)\right]\hfill \\ \hfill & -F\left(h_{min}\right){\Delta }_{\Omega }(c;m,n)\hfill \end{array}$$

(21)

or

$$\begin{array}{cc}\hfill F\left(\Omega \right(h\left)\right)& ⩽{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & -c\left[(m+n)F\left(h\right)-mn-F\left(h^2\right)\right]\hfill \\ \hfill & -F\left(h_{min}\right){\Delta }_{\Omega }(c;m,n).\hfill \end{array}$$

(22)

Finally, we mention another important result from the paper [5]. For $\Omega \in S{C}_c\left(\mathbb{I}\right),$ where $\mathbb{I}\subseteq \mathbb{R}$ is an open interval, $\delta ,\rho \ge 0,$$\delta +\rho =1,$ and $m,n\in \mathbb{I},$$m<n,$ we have

$$\begin{array}{cc}\hfill 0& ⩽min\left\{\delta ,\rho \right\}\left[\Omega \left(m\right)+\Omega \left(n\right)-2\Omega \left({\displaystyle \frac{m+n}{2}}\right)-{\displaystyle \frac{c}{2}}{(m-n)}^2\right]\hfill \\ \hfill & ⩽\delta \Omega \left(m\right)+\rho \Omega \left(n\right)-\Omega \left(\delta m+\rho n\right)-c\left(\delta m^2+\rho n^2-{\left(\delta m+\rho n\right)}^2\right).\hfill \end{array}$$

(23)

In the following remark, we present some key derivations and conclusions that we use in the proofs of the new refinements.

Remark 3.

It holds

$$\begin{array}{cc}\hfill c\delta \rho {(n-m)}^2& =c\delta \rho n^2-2c\delta \rho nm+c\delta \rho m^2\hfill \\ \hfill & =c\left(\delta m^2+\rho n^2-{\left(\delta m+\rho n\right)}^2\right)\hfill \\ \hfill & =c\left(\delta m^2+\rho n^2-{\delta }^2{m}^2-2\delta \rho mn-{\rho }^2{n}^2\right)\hfill \\ \hfill & =c\left(\delta m^2(1-\delta )+\rho n^2(1-\rho )-2\delta \rho mn\right)\hfill \\ \hfill & =c\left(\delta \rho m^2+\delta \rho n^2-2\delta \rho mn\right),\hfill \end{array}$$

i.e. inequality (23) is equivalent to

$$\begin{array}{cc}\hfill & \Omega \left(\delta m+\rho n\right)\hfill \\ \hfill & ⩽\delta \Omega \left(m\right)+\rho \Omega \left(n\right)-c\delta \rho {(n-m)}^2-min\left\{\delta ,\rho \right\}{\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & =\delta \Omega \left(m\right)+\rho \Omega \left(n\right)-c\left(\delta m^2+\rho n^2-{\left(\delta m+\rho n\right)}^2\right)\hfill \\ \hfill & -min\left\{\delta ,\rho \right\}{\Delta }_{\Omega }(c;m,n)\hfill \end{array}$$

(24)

Further we have

$$\begin{array}{cc}\hfill & c\delta \rho {(n-m)}^2+min\left\{\delta ,\rho \right\}{\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & =c\delta \rho {(n-m)}^2\hfill \\ \hfill & +min\left\{\delta ,\rho \right\}\left[\Omega \left(m\right)+\Omega \left(n\right)-2\Omega \left({\displaystyle \frac{m+n}{2}}\right)-{\displaystyle \frac{c}{2}}{\left(m-n\right)}^2\right]\hfill \\ \hfill & =c\delta \rho {(n-m)}^2-min\left\{\delta ,\rho \right\}{\displaystyle \frac{c}{2}}{\left(n-m\right)}^2\hfill \\ \hfill & +min\left\{\delta ,\rho \right\}\left[\Omega \left(m\right)+\Omega \left(n\right)-2\Omega \left({\displaystyle \frac{m+n}{2}}\right)\right]\hfill \\ \hfill & ⩾min\left\{\delta ,\rho \right\}\left[\Omega \left(m\right)+\Omega \left(n\right)-2\Omega \left({\displaystyle \frac{m+n}{2}}\right)\right]\hfill \\ \hfill & =min\left\{\delta ,\rho \right\}{\Delta }_{\Omega }(m,n),\hfill \end{array}$$

(25)

because of

$$\begin{array}{cc}\hfill & c\delta \rho {(n-m)}^2-min\left\{\delta ,\rho \right\}{\displaystyle \frac{c}{2}}{\left(n-m\right)}^2\hfill \\ \hfill & =\left\{\begin{array}{cc}\hfill \left[c\right]llc\delta {\left(n-m\right)}^2\left(\rho -{\displaystyle \frac{1}{2}}\right)\ge 0,& for0\le \delta \le {\displaystyle \frac{1}{2}},i.e.min\left\{\delta ,\rho \right\}=\delta \\ \hfill c\rho {(n-m)}^2\left(\delta -{\displaystyle \frac{1}{2}}\right)>0,& for{\displaystyle \frac{1}{2}}<\delta \le 1,i.e.min\left\{\delta ,\rho \right\}=\rho \end{array}\right.\hfill \end{array}$$

The growing number of recent studies highlights the significant role and wide applicability of Jensen’s inequality in various areas of mathematics and optimization. In 2009, M. A. Khan [6] developed improved forms of Jensen’s inequality for convex and monotone functions. Their work also included several applications to different types of means, and they further investigated related Jensen-type inequalities along with applications involving the Cauchy mean. Jensen’s inequality has been improved in a number of ways to increase its range of applications and provide more accurate bounds. D. Choi [7] proposed a method based on linear interpolation to improve Jensen-type inequalities for convex and piecewise convex functions. Their results also produced refinements of Young-type and matrix inequalities. In 2015, Matković et al.[8] improved the Jessen-Mercer inequality, provided generalizations for convex functions, and introduced a method for constructing exponentially convex functions. In addition, several mean value theorems were established and applied to study Stolarsky-type means using Jessen-Mercer differences. The generalization of Jensen-type inequalities to broader convexity classes has resulted in new findings with significant applications in integral and functional analysis. In this direction, S. S. Dragomir (2018) [9] derived several Lebesgue integral Jensen-type inequalities for convex functions defined on real intervals, thereby further enriching the theory of integral inequalities in generalized convex settings. Operator variants of Jensen’s inequality have also received considerable attention. M. S. Hosseini [10] studied operator Jensen’s inequalities and established a new general form along with its reverse version for convex functions that are not necessarily operator convex. In 2023, S. I. Bradanović [5] used strongly convex functions to improve Jessen’s and Jensen’s inequalities, including their converses and interpolating forms. These improved inequalities were then applied to define and study strongly f-divergences. As a result, stronger bounds were obtained for well-known divergences such as Kullback-Leibler, ${\chi }^2,$ Hellinger, Bhattacharyya, and Jeffreys distances. In 2025, Dragomir et al. [11] presented key characterizations of the generalized $(m,M,\psi )$-convex functions and used these results to establish majorization-type inequalities, leading to improved estimates for several classical mean inequalities.

The main objective of this paper is to establish new refinements of the Jessen-Mercer inequality and its converse by employing the class of strongly convex functions. More precisely, recent improvements of the Jessen (14) and converse Jessen inequalities (15), together with the newly established structural characterization of strong convexity (24), enable us to derive sharper Jessen-Mercer type inequalities that improve inequalities (11) and (12). Furthermore, our main results lead to improved integral and probabilistic inequalities of Jensen-Mercer type, as well as to inequalities involving generalized logarithmic means and their special cases.

Our exposition is structured as follows. In Section 2, we establish the main results concerning new refinements of the Jessen-Mercer inequality and its converse for strongly convex functions. In doing so, we also clarify how these results relate to the classical Jensen framework. Section 3 is devoted to applications in the integral setting, where we derive inequalities improving several recent results from [12]. As particular cases, we obtain new estimates for generalized logarithmic means. Finally, in the last section, we formulate probabilistic counterparts of the obtained inequalities. These results refine and extend several previously published inequalities from [4,13] and [14].

2. Main Results

Our first statement can be viewed as an improved version of the reflected Jessen inequality (11). It is obtained by applying the Jessen inequality for strongly convex functions (14) to the reflected mapping $h\mapsto m+n-h.$ The reflected form makes it possible to derive estimates which differ from those in the classical setting and, in some cases, provide more accurate bounds. The influence of strong convexity is reflected through the additional quadratic term appearing in the inequalities.

Throughout the remainder of the paper, $\mathbb{I}\subseteq \mathbb{R}$ denotes an open interval, unless stated otherwise.

Theorem 1.

Let $\Omega \in S{C}_c\left(\mathbb{I}\right),$$m,n\in \mathbb{I}$ with $m<n$ and $0⩽K⩽1.$ Further, let $F\in \mathcal{L}(L,\mathbb{R}),$$h\in L$ takes values in $[m,n]$ and assume $h^2,\Omega (m+n-h)\in L.$ Then

$$\begin{array}{cc}\hfill & \Omega \left(m+n-F\left(h\right)\right)\hfill \\ \hfill & ⩽\Omega \left(m+n-F\left(h\right)\right)\hfill \\ \hfill & +K\left[F\left(\Omega (m+n-h)\right)-\Omega \left(m+n-F\left(h\right)\right)-c\left(F\left(h^2\right)-{\left(F\left(h\right)\right)}^2\right)\right]\hfill \\ \hfill & ⩽F\left(\Omega (m+n-h)\right)-c\left(F\left(h^2\right)-{\left(F\left(h\right)\right)}^2\right)\hfill \\ \hfill & ⩽F\left(\Omega (m+n-h)\right).\hfill \end{array}$$

(26)

Proof. 

By applying (14) to $m+n-h$ instead of $h,$ we get

$$\begin{array}{cc}\hfill & \Omega \left(F\right(m+n-h\left)\right)\hfill \\ \hfill & ⩽\Omega (m+n-F(h\left)\right)\hfill \\ \hfill & ⩽\Omega (m+n-F(h\left)\right)\hfill \\ \hfill & +m\left[F\left(\Omega (m+n-h)\right)-\Omega (m+n-F\left(h\right))-c\left[F\left({(m+n-h)}^2\right)-{\left(F(m+n-h)\right)}^2\right]\right]\hfill \\ \hfill & ⩽F\left(\Omega (m+n-h)\right)-c\left[F\left({(m+n-h)}^2\right)-{\left(F(m+n-h)\right)}^2\right]\hfill \\ \hfill & ⩽F\left(\Omega \right(m+n-h\left)\right).\hfill \end{array}$$

Since

$$F\left({(m+n-h)}^2\right)-{\left(F(m+n-h)\right)}^2=F\left(h^2\right)-{\left(F\left(h\right)\right)}^2,$$

the required result holds. □

Combining inequality (26) with the following auxiliary result, which gives several successive upper estimates for $F\left(\Omega (m+n-h)\right)$, leads to a refined version of inequality (11). In this way, one can observe how the obtained bounds are related to each other and how they can be gradually improved. This also gives a better overview of the structure of the corresponding inequalities.

Theorem 2.

Let $\Omega \in S{C}_c\left(\mathbb{I}\right),$ where $m,n\in \mathbb{I}$ satisfy $m<n.$ Assume that L is a lattice and $F\in \mathcal{L}(L,\mathbb{R}).$ Moreover, let $h\in L$ takes values in $[m,n]$ and suppose that $h^2,\Omega \left(h\right),\Omega (m+n-h)\in L$. Then the following inequalities hold:

$$\begin{array}{cc}\hfill & F\left(\Omega (m+n-h)\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)-c\left(F\left(h\right)-m\right)\left(n-F\left(h\right)\right)-F\left(h_{min}\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-c{R}_{F\left(h\right)}(m,n)-F\left(h_{min}\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-c{R}_{F\left(h\right)}(m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \right(h\left)\right),\hfill \end{array}$$

(27)

where $h_{min},$$R_{F\left(h\right)}(m,n),$ and ${\Delta }_{\Omega }(c;m,n)$ are defined by (16), (18) and (19), respectively.

Proof. 

Define the mappings $\delta ,\rho :\mathbb{I}\to \mathbb{R}$ by

$$\delta \left(x\right)={\displaystyle \frac{n-x}{n-m}},\rho \left(x\right)={\displaystyle \frac{x-m}{n-m}}.$$

(28)

Then for every $x\in \mathbb{I},$

$$\Omega \left(x\right)=\Omega \left({\displaystyle \frac{n-x}{n-m}}m+{\displaystyle \frac{x-m}{n-m}}n\right)=\Omega \left(\delta \left(x\right)m+\rho \left(x\right)n\right).$$

(29)

Implementing (29) to (24), we get

$$\begin{array}{cc}\hfill \Omega \left(x\right)& =\Omega \left(\delta \right(x)m+\rho (x\left)n\right)\hfill \\ \hfill & \le \delta \left(x\right)\Omega \left(m\right)+\rho \left(x\right)\Omega \left(n\right)-c\delta \left(x\right)\rho \left(x\right){\left(n-m\right)}^2\hfill \\ \hfill & -min\{\delta \left(x\right),\rho \left(x\right)\}{\Delta }_{\Omega }(c;m,n),\hfill \end{array}$$

i.e.

$$\begin{array}{cc}\hfill \Omega \left(x\right)& ⩽{\displaystyle \frac{n-x}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{x-m}{n-m}}\Omega \left(n\right)-c(n-x)(x-m)\hfill \\ \hfill & -min\{\delta ,\rho \}{\Delta }_{\Omega }(c;m,n).\hfill \end{array}$$

(30)

Now for $h\in L,$ such that $\Omega \left(h\right)\in L$ and $\Omega (m+n-h)\in L,$ first substituting $m+n-h$ in place of $x,$ and then applying F on both sides of (30), with $min\{\delta ,\rho \}={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}|h-{\displaystyle \frac{m+n}{2}}|=h_{min}$, we obtain

$$\begin{array}{cc}\hfill & F\left(\Omega (m+n-h)\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)-c\left(F\left(h\right)-m\right)\left(n-F\left(h\right)\right)-F\left(h_{min}\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & =\Omega \left(m\right)+\Omega \left(n\right)+\left({\displaystyle \frac{F\left(h\right)-m}{n-m}}-1\right)\Omega \left(m\right)+\left({\displaystyle \frac{n-F\left(h\right)}{n-m}}-1\right)\Omega \left(n\right)\hfill \\ \hfill & =\Omega \left(m\right)+\Omega \left(n\right)-\left[{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)\right]\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-c{R}_{F\left(h\right)}(m,n)-F\left(h_{min}\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-c{R}_{F\left(h\right)}(m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \right(h\left)\right)\hfill \end{array}$$

where the final inequalities are obtained from the observation that

$$\begin{array}{cc}\hfill -\left[{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)\right]& ⩽-F\left(\Omega \left(h\right)\right)-c{R}_{F\left(h\right)}(m,n)\hfill \\ \hfill & -F\left(h_{min}\right){\Delta }_{\Omega }(c;m,n),\hfill \end{array}$$

which is consequence of (15) and from the non negativity of the terms $R_{F\left(h\right)}(m,n)$ and ${\Delta }_{\Omega }(c;m,n).$ □

The following theorem presents an additional improvement of the Jessen-Mercer type inequality (12). More precisely, the result introduces a chain of intermediate inequalities that further clarify the relationship between the involved expressions and yield tighter estimates. This also highlights how intermediate steps can be used to better track the behavior of the corresponding bounds.

Theorem 3.

Let $\Omega \in S{C}_c\left(\mathbb{I}\right),$ where $m,n\in \mathbb{I}$ satisfy $m<n.$ Assume that L is a lattice and $F\in \mathcal{L}(L,\mathbb{R}).$ Moreover, let $h\in L$ takes values in $[m,n]$ and suppose that $\Omega \left(h\right)\in L$. Then the following inequalities hold:

$$\begin{array}{cc}\hfill & \Omega (m+n-F(h\left)\right)\hfill \\ \hfill & \le {\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & -c\left[{\displaystyle \frac{F\left(h\right)-m}{n-m}}{m}^2+{\displaystyle \frac{n-F\left(h\right)}{n-m}}{n}^2-{\left({\displaystyle \frac{F\left(h\right)-m}{n-m}}m+{\displaystyle \frac{n-F\left(h\right)}{n-m}}n\right)}^2\right]\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & \le {\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)\hfill \\ \hfill & -\left[1-{\displaystyle \frac{1}{n-m}}\left(F\left(\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)+\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right)\right]{\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)\hfill \\ \hfill & -\left[1-{\displaystyle \frac{2}{n-m}}F\left(\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)\right]{\Delta }_{\Omega }(m,n)\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right),\hfill \end{array}$$

(32)

where ${\Delta }_{\Omega }(m,n)$ and ${\Delta }_{\Omega }(c;m,n)$ are defined by (13) and (19), respectively.

Proof.We again consider the functions $\delta ,\rho :\mathbb{I}\to \mathbb{R}$ introduced in (28). Applying inequality (29) to relation (24), together with the second representation and estimate given in Remark 3, yields

$$\begin{array}{cc}\hfill \Omega \left(x\right)& \le \delta \left(x\right)\Omega \left(m\right)+\rho \left(x\right)\Omega \left(n\right)-c\left(\delta \left(x\right)m^2+\rho \left(x\right)n^2-{\left(\delta \left(x\right)m+\rho \left(x\right)n\right)}^2\right)\hfill \\ \hfill & -min\left\{\delta \left(x\right),\rho \left(x\right)\right\}{\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & \le \delta \left(x\right)\Omega \left(m\right)+\rho \left(x\right)\Omega \left(n\right)-min\left\{\delta \left(x\right),\rho \left(x\right)\right\}{\Delta }_{\Omega }(m,n).\hfill \end{array}$$

By substituting $x=F\left(h\right),$ we have

$$min\left\{\delta \left(F\right(h\left)\right),\rho \left(F\right(h\left)\right)\right\}=min\left\{{\displaystyle \frac{n-F\left(h\right)}{n-m}},{\displaystyle \frac{F\left(h\right)-m}{n-m}}\right\}={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{n+m}{2}}\right|$$

and

$$\begin{array}{cc}\hfill \Omega \left(F\right(h\left)\right)& \le {\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & -c\left[{\displaystyle \frac{n-F\left(h\right)}{n-m}}{m}^2+{\displaystyle \frac{F\left(h\right)-m}{n-m}}{n}^2-{\left({\displaystyle \frac{n-F\left(h\right)}{n-m}}m+{\displaystyle \frac{F\left(h\right)-m}{n-m}}n\right)}^2\right]\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & \le {\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(m,n).\hfill \end{array}$$

Replacing h by $m+n-h$ further gives

$$\begin{array}{cc}\hfill & \Omega (m+n-F(h\left)\right)\hfill \\ \hfill & \le {\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & -c\left[{\displaystyle \frac{F\left(h\right)-m}{n-m}}{m}^2+{\displaystyle \frac{n-F\left(h\right)}{n-m}}{n}^2-{\left({\displaystyle \frac{F\left(h\right)-m}{n-m}}m+{\displaystyle \frac{n-F\left(h\right)}{n-m}}n\right)}^2\right]\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & \le {\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(m,n)\hfill \\ \hfill & =\Omega \left(m\right)+\Omega \left(n\right)-\left[{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)\right]\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-F\left(h_{min}\right){\Delta }_{\Omega }(m,n)\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(m,n).\hfill \end{array}$$

(33)

The last inequality follows directly from (20), namely from the estimate

$$-\left[{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)\right]⩽-F\left(\Omega \left(h\right)\right)-F\left(h_{min}\right){\Delta }_{\Omega }(m,n),$$

where $F\left(h_{min}\right)$ is defined by (17).

Further, we have

$$\begin{array}{cc}\hfill & \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-F\left(h_{min}\right){\Delta }_{\Omega }(m,n)\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(m,n)\hfill \\ \hfill & =\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)\hfill \\ \hfill & -\left(1-{\displaystyle \frac{1}{n-m}}\left(F\left(\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)+\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right)\right){\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-\left(1-{\displaystyle \frac{2}{n-m}}F\left(\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)\right){\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right),\hfill \end{array}$$

(34)

where we use the fact

$$\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|=\left|F\left(h-{\displaystyle \frac{m+n}{2}}\right)\right|\le F\left|h-{\displaystyle \frac{m+n}{2}}\right|.$$

Now, combining (33) and (34), we get the required result (31). □

Finally, we provide an alternative formulation of the refined Jessen-Mercer inequality in the setting of strongly convex functions. This version emphasizes a different representation of the obtained bounds and complements the previous refinements discussed in this section. In particular, it shows that the same inequality can be expressed in several equivalent forms depending on the chosen approach.

Theorem 4.

Let $\Omega \in S{C}_c\left(\mathbb{I}\right),$ where $m,n\in \mathbb{I}$ satisfy $m<n.$ Assume that L is a lattice and $F\in \mathcal{L}(L,\mathbb{R}).$ Moreover, let $h\in L$ takes values in $[m,n]$ and suppose that $h^2,\Omega \left(h\right)\in L$. Then the following inequalities hold:

$$\begin{array}{cc}\hfill & \Omega (m+n-F(h\left)\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)-c\left[(m+n)F\left(h\right)-mn-{\left(F\left(h\right)\right)}^2\right]\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-c\left[2(m+n)F\left(h\right)-2mn-{\left(F\left(h\right)\right)}^2-F\left(h^2\right)\right]\hfill \\ \hfill & -\left[1-{\displaystyle \frac{1}{n-m}}\left(\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|+F\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)\right]{\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-c\left[2(m+n)F\left(h\right)-2mn-{\left(F\left(h\right)\right)}^2-F\left(h^2\right)\right]\hfill \\ \hfill & -\left(1-{\displaystyle \frac{2}{n-m}}F\left|h-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \left(h\right)\right)-c\left[2(m+n)F\left(h\right)-2mn-{\left(F\left(h\right)\right)}^2-F\left(h^2\right)\right]\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \right(h\left)\right),\hfill \end{array}$$

(35)

where ${\Delta }_{\Omega }(c;m,n)$ is defined by (19).

Proof. 

Replacing x with $F\left(h\right)$ in (30) gives

$$\begin{array}{cc}\hfill \Omega \left(F\left(h\right)\right)& ⩽{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)-c(n-F\left(h\right))(F\left(h\right)-m)\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n).\hfill \end{array}$$

After substituting $h\mapsto m+n-h,$ we obtain

$$\begin{array}{cc}\hfill & \Omega (m+n-F(h\left)\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right))-c\left[(m+n)F\left(h\right)-mn-{\left(F\left(h\right)\right)}^2\right]\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n).\hfill \end{array}$$

Next, we add and subtract the quantities $\Omega \left(m\right)$ and $\Omega \left(n\right),$

$$\begin{array}{cc}\hfill & \Omega (m+n-F(h\left)\right)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)-\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)-\Omega \left(n\right)\hfill \\ \hfill & -c\left[(m+n)F\left(h\right)-mn-{\left(F\left(h\right)\right)}^2\right]-\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-\left[{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)\right]\hfill \\ \hfill & -c\left[(m+n)F\left(h\right)-mn-{\left(F\left(h\right)\right)}^2\right]-\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n),\hfill \end{array}$$

(36)

Applying the functional F to both sides of (30) leads to

$$\begin{array}{cc}\hfill & F\left(\Omega \left(h\right)\right)⩽{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)-c\left[(m+n)F\left(h\right)-mn-F\left(h^2\right)\right]\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}F\left|h-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n).\hfill \end{array}$$

Consequently, we arrive at

$$\begin{array}{cc}\hfill & -F\left(\Omega \left(h\right)\right)-c\left((m+n)F\left(h\right)-mn-F\left(h^2\right)\right)-\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}F\left|h-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩾-\left[{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(n\right)\right].\hfill \end{array}$$

(37)

Inserting inequality (37) into (36) finally yields

$$\begin{array}{cc}\hfill & \Omega (m+n-F(h\left)\right)\hfill \\ \hfill & ⩽{\displaystyle \frac{F\left(h\right)-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-F\left(h\right)}{n-m}}\Omega \left(n\right)-c\left[(m+n)F\left(h\right)-mn-{\left(F\left(h\right)\right)}^2\right]\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-F\left(\Omega \right(h\left)\right)\hfill \\ \hfill & -c\left(2(m+n)F\left(h\right)-2mn-F\left(h^2\right)-{\left(F\left(h\right)\right)}^2\right)\hfill \\ \hfill & -\left(1-{\displaystyle \frac{1}{n-m}}\left(F\left|h-{\displaystyle \frac{m+n}{2}}\right|+\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|\right)\right){\Delta }_{\Omega }(c;m,n).\hfill \end{array}$$

(38)

Using additionally

$$\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|=\left|F\left(h-{\displaystyle \frac{m+n}{2}}\right)\right|\le F\left|h-{\displaystyle \frac{m+n}{2}}\right|,$$

we derive

$$\begin{array}{cc}\hfill & -\left(1-{\displaystyle \frac{1}{n-m}}\left(\left|F\left(h\right)-{\displaystyle \frac{m+n}{2}}\right|+F\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽-\left(1-{\displaystyle \frac{2}{n-m}}F\left|h-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n).\hfill \end{array}$$

Combining this relation with the non negativity of the quantities

$$\left(1-{\displaystyle \frac{2}{n-m}}F\left|h-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)$$

and

$$\begin{array}{cc}\hfill & c\left[2(m+n)F\left(h\right)-2mn-F\left(h^2\right)-{\left(F\left(h\right)\right)}^2\right]\hfill \\ \hfill & =c\left[2(m+n)F\left(h\right)-2mn-2F\left(h^2\right)+F\left(h^2\right)-{\left(F\left(h\right)\right)}^2\right]\hfill \\ \hfill & =c\left[2F\left((h-m)(n-h)\right)+F\left(h^2\right)-{\left(F\left(h\right)\right)}^2\right]\ge 0,\hfill \end{array}$$

immediately implies inequality (35). Finally, the proof is complete. □

3. Applications to Integral Inequalities and Generalized Logarithmic Means

In this section, we derive new integral inequalities for strongly convex functions using the previous main results. Since the functional setting based on positive linear functionals provides a general setting, we now pass to its integral version, which can be viewed as a natural realization in terms of a probability measure space. In this setting, the abstract functional F is represented by an integral functional. In the following results we assume that $\left(T,\Sigma ,\mu \right)$ is a probability measure space, $L=L_1\left(\mu \right)$ and for $h\in L_1\left(\mu \right),$ we define a linear, positive and normalized functional F given by

$$F\left(h\right)={\int }_{T}h\left(t\right)d\mu .$$

As direct consequence of Theorem 1, we obtain the following result.

Corollary 1.

Let $\Omega \in S{C}_c\left(\mathbb{I}\right),$ where $m,n\in \mathbb{I}$ satisfy $m<n$ and $0⩽K⩽1.$ Assume that $h\in L_1\left(\mu \right)$ takes values in $[m,n]$ and suppose that $h^2,\Omega (m+n-h)\in L_1\left(\mu \right)$. Then the following inequalities hold:

$$\begin{array}{cc}\hfill & \Omega \left(m+n-\overline{h}\right)\hfill \\ \hfill & ⩽\Omega \left(m+n-\overline{h}\right)\hfill \\ \hfill & +K\left[{\int }_T\Omega \left(m+n-h\left(t\right)\right)d\mu -\Omega \left(m+n-\overline{h}\right)-c\left({\int }_T{h}^2\left(t\right)d\mu -{\overline{h}}^2\right)\right]\hfill \\ \hfill & ⩽{\int }_T\Omega \left(m+n-h\left(t\right)\right)d\mu -c\left({\int }_T{h}^2\left(t\right)d\mu -{\overline{h}}^2\right)\hfill \\ \hfill & ⩽{\int }_T\Omega \left(m+n-h\left(t\right)\right)d\mu ,\hfill \end{array}$$

(39)

where $\overline{h}={\int }_{T}h\left(t\right)d\mu .$

The next statement can be deduced directly from Theorem 2.

Corollary 2.

Let $\Omega \in S{C}_c\left(\mathbb{I}\right),$ where $m,n\in \mathbb{I}$ satisfy $m<n$ and $0⩽K⩽1.$ Assume that $h\in L_1\left(\mu \right)$ takes values in $[m,n]$ and suppose that $h^2,\Omega \left(h\right),\Omega (m+n-h)\in L_1\left(\mu \right)$. Then the following inequalities hold:

$$\begin{array}{cc}\hfill & {\int }_T\Omega \left(m+n-h\left(t\right)\right)d\mu \hfill \\ \hfill & ⩽{\displaystyle \frac{\overline{h}-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-\overline{h}}{n-m}}\Omega \left(n\right)-c\left(\overline{h}-m\right)\left(n-\overline{h}\right)\hfill \\ \hfill & -\left[{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}{\int }_T\left|h-{\displaystyle \frac{m+n}{2}}\right|d\mu \right]{\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽{\displaystyle \frac{\overline{h}-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-\overline{h}}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-{\int }_T\Omega \left(h\left(t\right)\right)d\mu \hfill \\ \hfill & -c\left[{\displaystyle \frac{\overline{h}-m}{n-m}}{n}^2+{\displaystyle \frac{n-\overline{h}}{n-m}}{m}^2-{\int }_T{h}^2\left(t\right)d\mu \right]\hfill \\ \hfill & -\left[{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}{\int }_T\left|h-{\displaystyle \frac{m+n}{2}}\right|d\mu \right]{\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-{\int }_T\Omega \left(h\left(t\right)\right)d\mu \hfill \\ \hfill & -c\left[{\displaystyle \frac{\overline{h}-m}{n-m}}{n}^2+{\displaystyle \frac{n-\overline{h}}{n-m}}{m}^2-{\int }_T{h}^2\left(t\right)d\mu \right]\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-{\int }_T\Omega \left(h\left(t\right)\right)d\mu ,\hfill \end{array}$$

(40)

where $\overline{h}={\int }_{T}h\left(t\right)d\mu ,$ while ${\Delta }_{\Omega }(c;m,n)$ is introduced in 19).

The next corollary is direct consequence of Theorem 3.

Corollary 3.

Let $\Omega \in S{C}_c\left(\mathbb{I}\right),$ where $m,n\in \mathbb{I}$ satisfy $m<n$ and $0⩽K⩽1.$ Assume that $h\in L_1\left(\mu \right)$ takes values in $[m,n]$ and suppose that $\Omega \left(h\right)\in L_1\left(\mu \right)$. Then the following inequalities hold:

$$\begin{array}{cc}\hfill & \Omega (m+n-\overline{h})\hfill \\ \hfill & \le {\displaystyle \frac{\overline{h}-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-\overline{h}}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & -c\left[{\displaystyle \frac{\overline{h}-m}{n-m}}{m}^2+{\displaystyle \frac{n-\overline{h}}{n-m}}{n}^2-{\left({\displaystyle \frac{\overline{h}-m}{n-m}}m+{\displaystyle \frac{n-\overline{h}}{n-m}}n\right)}^2\right]\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|\overline{h}-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & \le {\displaystyle \frac{\overline{h}-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-\overline{h}}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|\overline{h}-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-{\int }_T\Omega \left(h\left(t\right)\right)d\mu \hfill \\ \hfill & -\left[1-{\displaystyle \frac{1}{n-m}}\left({\int }_T\left|h-{\displaystyle \frac{m+n}{2}}\right|d\mu -\left|\overline{h}-{\displaystyle \frac{m+n}{2}}\right|\right)\right]{\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-{\int }_T\Omega \left(h\left(t\right)\right)d\mu \hfill \\ \hfill & -\left[1-{\displaystyle \frac{2}{n-m}}{\int }_T\left|h-{\displaystyle \frac{m+n}{2}}\right|d\mu \right]{\Delta }_{\Omega }(m,n)\hfill \\ \hfill & \le \Omega \left(m\right)+\Omega \left(n\right)-{\int }_T\Omega \left(h\left(t\right)\right)d\mu ,\hfill \end{array}$$

(41)

where $\overline{h}={\int }_{T}h\left(t\right)d\mu$ and the quantities ${\Delta }_{\Omega }(m,n),{\Delta }_{\Omega }(c;m,n)$ are given by (13), (19), respectively.

Remark 4.

Our result (39) together with (41) improves result from ([12] Theorem 5.2.).

Finally, as a special case of Theorem 4, we get the following result.

Corollary 4.

Let $\Omega \in S{C}_c\left(\mathbb{I}\right),$ where $m,n\in \mathbb{I}$ satisfy $m<n$ and $0⩽K⩽1.$ Assume that $h\in L_1\left(\mu \right)$ takes values in $[m,n]$ and suppose that $h^2,\Omega \left(h\right)\in L_1\left(\mu \right)$. Then the following inequalities hold:

$$\begin{array}{cc}\hfill & \Omega (m+n-\overline{h})\hfill \\ \hfill & ⩽{\displaystyle \frac{\overline{h}-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-\overline{h}}{n-m}}\Omega \left(n\right)-c\left[(m+n)\overline{h}-mn-{\overline{h}}^2\right]\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\left|\overline{h}-{\displaystyle \frac{m+n}{2}}\right|\right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-{\int }_T\Omega \left(h\left(t\right)\right)d\mu -c\left[2(m+n)\overline{h}-2mn-{\overline{h}}^2-{\int }_T{h}^2\left(t\right)d\mu \right]\hfill \\ \hfill & -\left[1-{\displaystyle \frac{1}{n-m}}\left(\left|\overline{h}-{\displaystyle \frac{m+n}{2}}\right|+{\int }_T\left|h-{\displaystyle \frac{m+n}{2}}\right|d\mu \right)\right]{\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-{\int }_T\Omega \left(h\left(t\right)\right)d\mu -c\left[2(m+n)\overline{h}-2mn-{\overline{h}}^2-{\int }_T{h}^2\left(t\right)d\mu \right]\hfill \\ \hfill & -\left(1-{\displaystyle \frac{2}{n-m}}{\int }_T\left|h-{\displaystyle \frac{m+n}{2}}\right|d\mu \right){\Delta }_{\Omega }(c;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-{\int }_T\Omega \left(h\left(t\right)\right)d\mu \hfill \\ \hfill & -c\left[2(m+n)\overline{h}-2mn-{\overline{h}}^2-{\int }_T{h}^2\left(t\right)d\mu \right]\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-{\int }_T\Omega \left(h\left(t\right)\right)d\mu ,\hfill \end{array}$$

where $\overline{h}={\int }_{T}h\left(t\right)d\mu$ with ${\Delta }_{\Omega }(c;m,n)$ defined in (19).

By applying the previously derived corollaries to a particular family of positive normalized linear functionals, we obtain new inequalities involving generalized logarithmic means with parameter $q\in \mathbb{R}\setminus \{-1,0\}$ defined by

$$L_q(m,n)=\left\{\begin{array}{cc}m,& \mathrm{if}m=n,\\ {\left[{\displaystyle \frac{n^{q+1}-m^{q+1}}{(q+1)(n-m)}}\right]}^{{\displaystyle \frac{1}{q}}},& \mathrm{if}m\ne n.\end{array}\right.$$

This class of means provides a continuous bridge between different classical averages and plays an important role in various comparison results. In particular, the parameter q allows a smooth transition between different types of generalized means. For $q=1$, we recover the arithmetic mean

$$L_1(m,n)=A(m,n)={\displaystyle \frac{m+n}{2}}.$$

Let $\mathbb{I}\subseteq (0,\infty )$ be an open interval and let $m,n\in \mathbb{I}$ with $m<n.$ We consider the interval $T=[m,n]\subseteq (0,\infty )$ equipped with the uniform probability measure

$$d\mu ={\displaystyle \frac{1}{n-m}}dt.$$

This choice of measure ensures that the associated functional corresponds to the classical average value over the interval. In this setting, the associated positive normalized linear functional F is given by

$$F\left(h\right)={\displaystyle \frac{1}{n-m}}{\int }_m^{n}h\left(t\right)dt.$$

The next identities can be viewed as particular cases of the previously established main results, obtained by choosing specific functions and the corresponding positive normalized linear functional. In this way, the general theory developed in the previous sections directly leads to the following expressions.

Assuming that $q\in \mathbb{R}$ and $h,h^q\in L_1\left(\mu \right),$ we obtain

$$\begin{array}{cc}\hfill F\left(h\right)& ={\displaystyle \frac{1}{n-m}}{\int }_m^{n}tdt={\displaystyle \frac{m+n}{2}},\hfill \\ \hfill F\left(h^q\right)& ={\displaystyle \frac{1}{n-m}}{\int }_m^n{t}^{q}dt=L_q^q(m,n),\hfill \\ \hfill F\left(h_{min}\right)& ={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}F\left(\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)={\displaystyle \frac{1}{4}}\hfill \end{array}$$

and

$$R_{F\left(h\right)}(m,n)={\displaystyle \frac{F\left(h\right)-m}{n-m}}{n}^2+{\displaystyle \frac{n-F\left(h\right)}{n-m}}{m}^2-F\left(h^2\right)={\displaystyle \frac{{(n-m)}^2}{6}}.$$

We also have

$$F\left(\left|h-{\displaystyle \frac{m+n}{2}}\right|\right)={\displaystyle \frac{1}{n-m}}{\int }_m^n\left|t-{\displaystyle \frac{m+n}{2}}\right|dt={\displaystyle \frac{n-m}{4}},$$

and for $\Omega \left(t\right)=t^q,q\in \mathbb{R}$, we get

$$\begin{array}{cc}\hfill \Omega (m+n-F(h\left)\right)& ={\left({\displaystyle \frac{m+n}{2}}\right)}^q=A^q(m,n)\hfill \\ \hfill F\left({(m+n-h)}^q\right)& ={\displaystyle \frac{1}{n-m}}{\int }_m^n{(m+n-t)}^{q}dt={\displaystyle \frac{n^{q+1}-m^{q+1}}{(q+1)(n-m)}}=L_q^q(m,n)\hfill \\ \hfill F\left(\Omega \right(h\left)\right)& ={\displaystyle \frac{n^{q+1}-m^{q+1}}{(q+1)(n-m)}}=L_q^q(m,n)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill F\left(h^q\right)-{\left(F\left(h\right)\right)}^q& ={\displaystyle \frac{1}{n-m}}{\int }_m^n{t}^{q}dt-{\left({\displaystyle \frac{1}{n-m}}{\int }_m^{n}tdt\right)}^q\hfill \\ \hfill & =L_q^q(m,n)-F^q(m,n).\hfill \end{array}$$

Taking into account the above derivations, the following examples provide further applications of the main results, highlighting inequalities between means. These examples also show how the abstract inequalities can be interpreted in a more concrete analytical setting. In particular, they demonstrate how different choices of functions lead to known and meaningful relations between classical means.

Example 2.

Applying Corollary 1 to the function $\Omega \left(t\right)=t^q,$$q\in \mathbb{R}$, which is strongly convex with modulus

$$c={\displaystyle \frac{1}{2}}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2},$$

yields the following inequality

$$\begin{array}{cc}\hfill & A^q(m,n)\hfill \\ \hfill & ⩽A^q(m,n)\hfill \\ \hfill & +K\left[L_q^q(m,n)-A^q(m,n)-{\displaystyle \frac{1}{24}}{(n-m)}^{2}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2}\right]\hfill \\ \hfill & ⩽L_q^q(m,n)-{\displaystyle \frac{1}{24}}{(n-m)}^{2}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2}\hfill \\ \hfill & ⩽L_q^q(m,n).\hfill \end{array}$$

Example 3.

Using Corollary 2 for the particular choice $\Omega \left(t\right)=t^q,q\in \mathbb{R}$, here the modulus of strong convexity is

$$c={\displaystyle \frac{1}{2}}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2},$$

we obtain

$$\begin{array}{cc}\hfill & L_q^q(m,n)\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}A(m^q,n^q)+{\displaystyle \frac{1}{2}}{A}^q(m,n)-{\displaystyle \frac{1}{16}}{\left(n-m\right)}^{2}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2}\hfill \\ \hfill & ⩽A(m^q,n^q)\hfill \\ \hfill & ⩽{\displaystyle \frac{3}{2}}A(m^q,n^q)+{\displaystyle \frac{1}{2}}{A}^q(m,n)-L_q^q(m,n)-{\displaystyle \frac{1}{48}}{(n-m)}^{2}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2}\hfill \\ \hfill & ⩽2A(m^q,n^q)-L_q^q(m,n)-{\displaystyle \frac{1}{12}}{(n-m)}^{2}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2}\hfill \\ \hfill & ⩽2A(m^q,n^q)-L_q^q(m,n).\hfill \end{array}$$

Example 4.

By specializing Corollary 4 to the strongly convex function $\Omega \left(t\right)=t^q,q\in \mathbb{R}$, with modulus

$$c={\displaystyle \frac{1}{2}}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2},$$

we arrive at

$$\begin{array}{cc}\hfill & A^q(m,n)\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{2}}A(m^q,n^q)+{\displaystyle \frac{3}{2}}{A}^q(m,n)-L_q^q(m,n)-{\displaystyle \frac{1}{48}}{(n-m)}^{2}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2}\hfill \\ \hfill & ⩽A(m^q,n^q)+A^q(m,n)-L_q^q(m,n)-{\displaystyle \frac{1}{12}}{(n-m)}^{2}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2}\hfill \\ \hfill & ⩽2A(m^q,n^q)-L_q^q(m,n)-{\displaystyle \frac{5}{24}}{(n-m)}^{2}q(q-1)\underset{t\in [m,n]}{min}{t}^{q-2}\hfill \\ \hfill & ⩽2A(m^q,n^q)-L_q^q(m,n).\hfill \end{array}$$

4. Application to Probabilistic Inequalities

In this section, we provide a probabilistic perspective on the previously established results. Throughout the discussion, we work with real valued random variables defined on a probability space $\left(T,\Sigma ,\mu \right)$ and assume that all considered expectations are finite. For a random variable $Y:T\to \mathbb{R}$, the symbols $\mathbb{E}\left[Y\right]$ and ${\mathbb{D}}^2\left[Y\right]$ denote its expectation and variance, respectively. It is well known that every convex function $\Omega :\mathbb{I}\to \mathbb{R}$ satisfies the classical Jensen inequality

$$\Omega \left(\mathbb{E}\right[Y\left]\right)⩽\mathbb{E}\left[\Omega \right(Y\left)\right]$$

(42)

for any random variable Y whose values belong to the interval $\mathbb{I}$.

A strengthened form of inequality (42) was established in [14]. More precisely, if $\Omega :\mathbb{I}\to \mathbb{R}$, is strongly convex with modulus c, then the following estimate holds:

$$\Omega \left(\mathbb{E}\left[Y\right]\right)\le \mathbb{E}\left[\Omega \left(Y\right)\right]-{\mathbb{D}}^2\left[Y\right]⩽\mathbb{E}\left[\Omega \left(Y\right)\right].$$

(43)

This inequality shows that the variance term naturally appears as a measure of deviation between the two expressions. The probabilistic reformulation of our principal results yields further refinements of inequality (43), together with several related inequalities of Jensen-Mercer type. In this setting, the obtained bounds provide a more detailed description of the deviation between the expectation of a transformed random variable and the transform of its expectation.

Let $\mathbb{I}\subseteq \mathbb{R}$ be an open interval and $\Omega :\mathbb{I}\to \mathbb{R}$ be a strongly convex function with modulus $c>0.$ Furthermore, let $m,n\in \mathbb{I}$ satisfy $m<n$. For a random variable $Y:T\to [m,n]$, under the assumptions $Y^2,\Omega \left(Y\right),\Omega (m+n-Y)\in L^1\left(\mu \right),$ we consider a positive normalized linear functional F such that

$$\begin{array}{cc}\hfill F\left(Y\right)& =\mathbb{E}\left[Y\right],\hfill \\ \hfill F\left(Y^2\right)& =\mathbb{E}\left[Y^2\right],\hfill \\ \hfill F\left(\Omega \right(Y\left)\right)& =\mathbb{E}\left[\Omega \right(Y\left)\right],\hfill \\ \hfill F\left(\Omega \right(m+n-Y\left)\right)& =\mathbb{E}\left[\Omega (m+n-Y)\right]\hfill \end{array}$$

and

$$F\left(Y^2\right)-{\left(F\left(Y\right)\right)}^2=\mathbb{E}\left[Y^2\right]-{\left(\mathbb{E}\left[Y\right]\right)}^2={\mathbb{D}}^2\left[Y\right]$$

The next corollary follows immediately from Theorem 1 and represents a direct application of the preceding result.

Corollary 5.

Let $\Omega \in S{C}_c\left(\mathbb{I}\right)$, $m,n\in \mathbb{I}$ with $m<n$ and $0⩽K⩽1.$ Further let $Y:T\to [m,n]$ be a random variable such that $Y^2,\Omega (m+n-Y)\in L^1\left(\mu \right).$ Then

$$\begin{array}{cc}\hfill & \Omega \left(m+n-\mathbb{E}\left[Y\right]\right)\hfill \\ \hfill & ⩽\Omega \left(m+n-\mathbb{E}\left[Y\right]\right)\hfill \\ \hfill & +K\left(\mathbb{E}\left[\Omega (m+n-Y)\right]-\Omega \left(m+n-\mathbb{E}\left[Y\right]\right)-c{\mathbb{D}}^2\left[Y\right]\right)\hfill \\ \hfill & ⩽\mathbb{E}\left[\Omega (m+n-Y)\right]-c{\mathbb{D}}^2\left[Y\right]\hfill \\ \hfill & ⩽\mathbb{E}\left[\Omega (m+n-Y)\right].\hfill \end{array}$$

(44)

Remark 5.

In particular, replacing $m+n-Y$ by Z and $m+n-\mathbb{E}\left[Y\right]$ by $\mathbb{E}\left[Z\right]$ in (44), leads to the following refinement of inequality (43):

$$\begin{array}{cc}\hfill \Omega \left(\mathbb{E}\left[Z\right]\right)& ⩽\Omega \left(\mathbb{E}\left[Z\right]\right)+K\left(\mathbb{E}\left[\Omega \left(Z\right)\right]-\Omega \left(\mathbb{E}\left[Z\right]\right)-c{\mathbb{D}}^2\left[Z\right]\right)\hfill \\ \hfill & ⩽\mathbb{E}\left[\Omega \left(Z\right)\right]-c{\mathbb{D}}^2\left[Z\right]\hfill \\ \hfill & ⩽\mathbb{E}\left[\Omega \left(Z\right)\right].\hfill \end{array}$$

(45)

An immediate consequence of Theorem 2 is given in the following statement.

Corollary 6.

Let $\Omega \in S{C}_c\left(\mathbb{I}\right)$ and $m,n\in \mathbb{I}$ with $m<n.$ Assume that $Y:T\to [m,n]$ is a random variable satisfying $\Omega \left(Y\right),\Omega (m+n-Y)\in L^1\left(\mu \right).$ Then the following inequalities hold:

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\Omega \left(m+n-Y\right)\right]\hfill \\ \hfill & ⩽{\displaystyle \frac{\mathbb{E}\left[Y\right]-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-\mathbb{E}\left[Y\right]}{n-m}}\Omega \left(n\right)-c\left(\mathbb{E}\left[Y\right]-m\right)\left(n-\mathbb{E}\left[Y\right]\right)\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\mathbb{E}\left[\left|Y-{\displaystyle \frac{m+n}{2}}\right|\right]\right){\Delta }_c(\Omega ;m,n)\hfill \\ \hfill & ⩽{\displaystyle \frac{\mathbb{E}\left[Y\right]-m}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{n-\mathbb{E}\left[Y\right]}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-\mathbb{E}\left[\Omega \left(Y\right)\right]-c\left({\displaystyle \frac{\mathbb{E}\left[Y\right]-m}{n-m}}{n}^2+{\displaystyle \frac{n-\mathbb{E}\left[Y\right]}{n-m}}{m}^2-\mathbb{E}\left[Y^2\right]\right)\hfill \\ \hfill & -\left(1-{\displaystyle \frac{2}{n-m}}\mathbb{E}\left[\left|{\displaystyle \frac{m+n}{2}}-Y\right|\right]\right){\Delta }_c(\Omega ;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-\mathbb{E}\left[\Omega \left(Y\right)\right]-c\left({\displaystyle \frac{\mathbb{E}\left[Y\right]-m}{n-m}}{n}^2+{\displaystyle \frac{n-\mathbb{E}\left[Y\right]}{n-m}}{m}^2-\mathbb{E}\left[Y^2\right]\right)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-\mathbb{E}\left[\Omega \left(Y\right)\right],\hfill \end{array}$$

where ${\Delta }_{\Omega }(c;m,n)$ is defined by relation (19).

Remark 6.

By introducing the substitutions $m+n-Y=Z$ and $m+n-\mathbb{E}\left[Y\right]=\mathbb{E}\left[Z\right]$ into the previous inequality, we obtain the corresponding reformulated expression:

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\Omega \left(Z\right)\right]\hfill \\ \hfill & ⩽{\displaystyle \frac{n-\mathbb{E}\left[Z\right]}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{\mathbb{E}\left[Z\right]-m}{n-m}}\Omega \left(n\right)-c\left(\mathbb{E}\left[Z\right]-m\right)\left(n-\mathbb{E}\left[Z\right]\right)\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n-m}}\mathbb{E}\left[\left|Z-{\displaystyle \frac{m+n}{2}}\right|\right]\right){\Delta }_c(\Omega ;m,n)\hfill \\ \hfill & ⩽{\displaystyle \frac{n-\mathbb{E}\left[Z\right]}{n-m}}\Omega \left(m\right)+{\displaystyle \frac{\mathbb{E}\left[Z\right]-m}{n-m}}\Omega \left(n\right)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-\mathbb{E}\left[\Omega (m+n-Z)\right]-c\left({\displaystyle \frac{n-\mathbb{E}\left[Z\right]}{n-m}}{n}^2+{\displaystyle \frac{\mathbb{E}\left[Z\right]-m}{n-m}}{m}^2-\mathbb{E}\left[{(m+n-Z)}^2\right]\right)\hfill \\ \hfill & -\left(1-{\displaystyle \frac{2}{n-m}}\mathbb{E}\left[\left|{\displaystyle \frac{m+n}{2}}-Z\right|\right]\right){\Delta }_c(\Omega ;m,n)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-\mathbb{E}\left[\Omega (m+n-Z)\right]-c\left({\displaystyle \frac{n-\mathbb{E}\left[Z\right]}{n-m}}{n}^2+{\displaystyle \frac{\mathbb{E}\left[Z\right]-m}{n-m}}{m}^2-\mathbb{E}\left[{(m+n-Z)}^2\right]\right)\hfill \\ \hfill & ⩽\Omega \left(m\right)+\Omega \left(n\right)-\mathbb{E}\left[\Omega (m+n-Z)\right].\hfill \end{array}$$

(46)

Combining inequalities (45) and (46) yields a refinement of the results established in ([4] Corollary 7.) and ([13] Corollary 9.), leading to sharper bounds in the considered setting.