Non-Archimedean Smale Mean Value and Dubinin–Sugawa Dual Mean Value Problems

1. Introduction

In 1981, Steve Smale proved the following result using the Koebe 1/4-theorem [1].

Theorem 1.

[1] (Smale Mean Value Theorem) Let $P\left(z\right):=a_0+a_{1}z+a_2{z}^2+\cdots +a_n{z}^n$ be a polynomial of degree $n\ge 2$ over $\mathbb{C}$. If $z\in \mathbb{C}$ is not a critical point of P (i.e., $P^{\prime }\left(z\right)\ne 0$), then

$$\begin{array}{c}\hfill \underset{w\in \mathbb{C},P^{\prime }\left(w\right)=0}{min}\left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|\le 4|P^{\prime }\left(z\right)|.\end{array}$$

(1)

Note that Theorem 1 can also be stated as follows.

Theorem 2.

[1] (Smale Mean Value Theorem) Let $P\left(z\right):=a_0+a_{1}z+a_2{z}^2+\cdots +a_n{z}^n$ be a polynomial of degree $n\ge 2$ over $\mathbb{C}$. If $z\in \mathbb{C}$ is not a critical point of P, then there exists a critical point $w\in \mathbb{C}$ of P such that

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|\le 4|P^{\prime }\left(z\right)|.\end{array}$$

(2)

The proof of Theorem 2 can also be found in books [2,3]. Inequality (1) led Smale to ask the following problem. What is the best constant $4>c>0$ such that the Inequality (1) can be replaced by

$$\begin{array}{c}\hfill \underset{w\in \mathbb{C},P^{\prime }\left(w\right)=0}{min}\left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|\le c|P^{\prime }\left(z\right)|?\end{array}$$

(3)

Smale made the following conjecture with regard to Inequality (3), which remains open today.

Conjecture 3.

[1,4,5,6,7,8] (Smale Mean Value Conjecture) Let $P\left(z\right):=a_0+a_{1}z+a_2{z}^2+\cdots +a_n{z}^n$ be a polynomial of degree $n\ge 2$ over $\mathbb{C}$. If $z\in \mathbb{C}$ is not a critical point of P, then

$$\begin{array}{c}\hfill \underset{w\in \mathbb{C},P^{\prime }\left(w\right)=0}{min}\left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|\le 1|P^{\prime }\left(z\right)|\end{array}$$

or

$$\begin{array}{c}\hfill \underset{w\in \mathbb{C},P^{\prime }\left(w\right)=0}{min}\left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|\le {\displaystyle \frac{n-1}{n}}|P^{\prime }\left(z\right)|={\displaystyle \frac{deg\left(P\right)-1}{deg\left(P\right)}}|P^{\prime }\left(z\right)|.\end{array}$$

(4)

By taking polynomial $P\left(z\right):=z^n-nz\in \mathbb{C}\left[z\right]$, we see that the constant $(n-1)/n$ in inequality (4) cannot be improved further. Conjecture 3 also appears as Problem 1 in the Addenda after the list of 18 open problems of Smale in Mathematical Problems for the Next Century [5]. Now for a polynomial P of degree $n\ge 2$, define

$$\begin{array}{c}\hfill S\left(P\right):=\underset{z\in \mathbb{C},P^{\prime }\left(z\right)\ne 0}{sup}{\displaystyle \frac{1}{|P^{\prime }\left(z\right)|}}\underset{w\in \mathbb{C},P^{\prime }\left(w\right)=0}{min}\left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|\end{array}$$

and

$$\begin{array}{c}\hfill M:=\underset{P\mathrm{is}\mathrm{a}\mathrm{non}\mathrm{linear}\mathrm{polynomial}}{sup}S\left(P\right).\end{array}$$

Smale conjecture then asks whether $M=1$ or $S\left(P\right)={\displaystyle \frac{\mathrm{deg}\left(P\right)-1}{\mathrm{deg}\left(P\right)}}$? It is known that the Conjecture 3 holds for polynomials of degree 2, 3, 4, [2,3,9]. Through computer calculations, Smale conjecture is known to hold for polynomials of degree upto 10 [10].

Usually, the study of Smale mean value conjecture goes with the following conjecture which is known as normalization equivalence to Conjecture 3.

Conjecture 4.

[1,2,3] (Normalized Smale Mean Value Conjecture) Let P be a degree $n\ge 2$ polynomial over $\mathbb{C}$ which is normalized, i.e., $P\left(0\right)=0$ and $P^{\prime }\left(0\right)=1$. Then

$$\begin{array}{c}\hfill S_0\left(P\right):=\underset{w\in \mathbb{C},P^{\prime }\left(w\right)=0}{min}\left|{\displaystyle \frac{P\left(w\right)}{w}}\right|={\displaystyle \frac{deg\left(P\right)-1}{deg\left(P\right)}}\end{array}$$

or

$$\begin{array}{c}\hfill M_0:=\underset{Pis\; a\; non\; linear\; polynomial,P\left(0\right)=0,P^{\prime }\left(0\right)=1}{sup}{S}_0\left(P\right)=1.\end{array}$$

In 1996, Andrievskii and Ruscheweyh showed that if critical points of a normalized polynomial P lie on the unit circle, then $S_0\left(P\right)\le 1$ [11]. In 2009, Hinkkanen and Kayumov showed that if all the critical points of a normalized polynomial P are real, then $S_0\left(P\right)\le {\displaystyle \frac{2}{3}}$ [12]. In 2010, Hinkkanen and Kayumov proved that if the critical points of normalized polynomial P lie on the sector $\{r{e}^{i\theta }:r>0,|\theta |\le \pi /6\}$, then $S_0\left(P\right)\le {\displaystyle \frac{1}{2}}$ and if the critical points of normalized polynomial P lie in the union of two rays $\{1+r{e}^{\pm i\theta }:r\ge 0\}$ for each $0<\theta \le \pi /2$, then $S_0\left(P\right)\le {\displaystyle \frac{2}{3}}$ [13]. In [2] it is showed that if the critical points of a normalized polynomial are all real, then $S_0\left(P\right)\le {\displaystyle \frac{\mathrm{deg}\left(P\right)-2}{\mathrm{deg}\left(P\right)}}\left[{\left({\displaystyle \frac{\mathrm{deg}\left(P\right)-1}{\mathrm{deg}\left(P\right)-2}}\right)}^{\mathrm{deg}\left(P\right)-1}-2\right]$ and in [3] it is showed that $S_0\left(P\right)\le e-2$.

In 2002, Beardon, Minda and Ng showed that $S\left(P\right)\le 4^{{\displaystyle \frac{\mathrm{deg}\left(P\right)-2}{\mathrm{deg}\left(P\right)-1}}}$ [14]. In 2006, Fijikawa and Sugawa proved that $S\left(P\right)\le 4{\displaystyle \frac{1+(\mathrm{deg}\left(P\right)-2)4^{\frac{-1}{\mathrm{deg}\left(P\right)-1}}}{\mathrm{deg}\left(P\right)+1}}$ [15]. In 2007, Conte, Fujikawa and Lakic showed that $S\left(P\right)\le 4{\displaystyle \frac{\mathrm{deg}\left(P\right)-1}{\mathrm{deg}\left(P\right)+1}}$ [16]. In 2007, Crane showed that there exists $n_0\ge 2$ such that for all $n\ge n_0$, $S\left(P\right)\le 4-{\displaystyle \frac{2}{\sqrt{\mathrm{deg}\left(P\right)}}}$ [17] (also see [18]). In 2003, Ng proved that Conjecture 3 holds with $M=2$ for all odd polynomials with nonzero linear terms [19]. In 2006, Dubinin proved that if the critical points of a normalized polynomial P have equal modulus, then $S_0\left(P\right)\le {\displaystyle \frac{deg\left(P\right)-1}{deg\left(P\right)}}$ [20].

At the other side, in 2002, Sendov formulated a generalization of Conjecture 3 [21]. In 2003, Sendov and Nikolov gave some variations of Conjecture 3 [22]. In 2005, Tyson showed that Tischler’s strong form of Smale mean value conjecture [9,23] fails [24]. One can interpret Conjecture 3 in the form of finite Blaschke products [3,25,26]. It is also possible to connect Conjecture 3 with electrostatics [27].

In 1981, Smale also derived the following higher order mean value theorem [1].

Theorem 5.

[1] (Higher Order Smale Mean Value Theorem) Let $P\left(z\right):=a_0+a_{1}z+a_2{z}^2+\cdots +a_n{z}^n$ be a polynomial of degree $n\ge 2$ over $\mathbb{C}$. If $z\in \mathbb{C}$ is not a critical point of P, then there exists a critical point $w\in \mathbb{C}$ of P such that

$$\begin{array}{c}\hfill {\displaystyle \frac{|P^{\left(k\right)}\left(z\right)|}{k!}}{\displaystyle \frac{{|P\left(z\right)-P\left(w\right)|}^{k-1}}{|P^{\prime }{\left(z\right)|}^k}}\le 4^{k-1},\forall 2\le k\le n.\end{array}$$

(5)

In 2009, Dubinin and Sugawa derived the following striking result which is a dual result for Smale mean value theorem [28].

Theorem 6.

[28,29] (Dubinin-Sugawa Dual Mean Value Theorem) Let $P\left(z\right):=a_0+a_{1}z+a_2{z}^2+\cdots +a_n{z}^n$ be a polynomial of degree $n\ge 2$ over $\mathbb{C}$. If $z\in \mathbb{C}$ is not a critical point of P, then there exists a critical point $w\in \mathbb{C}$ of P such that

$$\begin{array}{c}\hfill {\displaystyle \frac{|P^{\prime }\left(z\right)|}{n{4}^n}}\le \left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|.\end{array}$$

In other words,

$$\begin{array}{c}\hfill {\displaystyle \frac{|P^{\prime }\left(z\right)|}{n{4}^n}}\le \underset{w\in \mathbb{C},P^{\prime }\left(w\right)=0}{max}\left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|.\end{array}$$

Dubinin and Sugawa made the following conjecture in 2009 [28].

Conjecture 7.

[28,29] (Dubinin-Sugawa Dual Mean Value Conjecture) Let $P\left(z\right):=a_0+a_{1}z+a_2{z}^2+\cdots +a_n{z}^n$ be a polynomial of degree $n\ge 2$ over $\mathbb{C}$. If $z\in \mathbb{C}$ is not a critical point of P, then

$$\begin{array}{c}\hfill {\displaystyle \frac{|P^{\prime }\left(z\right)|}{n}}\le \underset{w\in \mathbb{C},P^{\prime }\left(w\right)=0}{max}\left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|.\end{array}$$

For a polynomial P of degree $n\ge 2$, define

$$\begin{array}{c}\hfill DS\left(P\right):=\underset{z\in \mathbb{C},P^{\prime }\left(z\right)\ne 0}{inf}{\displaystyle \frac{1}{|P^{\prime }\left(z\right)|}}\underset{w\in \mathbb{C},P^{\prime }\left(w\right)=0}{max}\left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|.\end{array}$$

Similar to Conjecture 4, we have the following normalized version of Conjecture 7.

Conjecture 8.

[28,29] (Normalized Smale Dubinin-Sugawa Dual Mean Value Conjecture) Let P be a degree $n\ge 2$ polynomial over $\mathbb{C}$ which is normalized, i.e., $P\left(0\right)=0$ and $P^{\prime }\left(0\right)=1$. Then

$$\begin{array}{c}\hfill D{S}_0\left(P\right):=\underset{w\in \mathbb{C},P^{\prime }\left(w\right)=0}{max}\left|{\displaystyle \frac{P\left(w\right)}{w}}\right|={\displaystyle \frac{1}{n}}={\displaystyle \frac{1}{deg\left(P\right)}}.\end{array}$$

In 2016, Ng and Zhang showed that if P is a normalized polynomial, then $D{S}_0\left(P\right)>{\displaystyle \frac{1}{4^{\mathrm{deg}}\left(\mathrm{P}\right)}}$ [26]. In 2019 Dubinin showed that $DS\left(P\right)\ge {\displaystyle \frac{1}{n}}tan\left({\displaystyle \frac{\pi }{4\mathrm{deg}\left(\mathrm{P}\right)}}\right)$ [30]. In 2019, Hinkkanen, Kayumov and Khammatova showed that Conjecture 7 holds for polynomials of degree six [31]. In 2025, Hinkkanen, Kayumov and Khammatova showed that Conjecture 7 holds for polynomials of degree six [32].

In this paper, we ask for non-Archimedean versions of Theorems 1 and 6. We show that if universal constants exist for the non-Archimedean Smale and the Dubinin–Sugawa dual mean value problems, the former is bounded below by unity, whereas the latter is bounded above by unity.

2. Non-Archimedean Smale Mean Value Problem

Let $\mathbb{K}$ be a field. Recall that a map $|·|:\mathbb{K}\to [0,\infty )$ is said to be a non-Archimedean valuation if the following conditions hold:

(i)

If $\lambda \in \mathbb{K}$ satisfies $|\lambda |=0$, then $\lambda =0$.

(ii)

$|\lambda \mu |=|\lambda ||\mu |$ for all $\lambda ,\mu \in \mathbb{K}$.

(iii)

(Ultratriangle inequality) $|\lambda +\mu |\le max\{\lambda |,|\mu |\}$ for all $\lambda ,\mu \in \mathbb{K}$.

We say $\mathbb{K}$ is non-Archimedean complete valued field if it is complete w.r.t. the ultrametric $d(\lambda ,\mu ):=|\lambda -\mu |$ for all $\lambda ,\mu \in \mathbb{K}$ [33]. Throughout the article we assume that $\mathbb{K}$ is algebraically closed.

We ask for non-Archimedean version of Theorem 2 as follows.

Problem 1.

(Non-Archimedean Smale Mean Value Problem) Let $\mathbb{K}$ be a non-Archimedean (complete) valued field. Whether there is a universal constant $S_{\mathbb{K}}$ (which may depend upon $\mathbb{K}$) satisfying the following: Let $P\left(z\right):=a_0+a_{1}z+a_2{z}^2+\cdots +a_n{z}^n$ be a polynomial of degree $n\ge 2$ over $\mathbb{K}$. If $z\in \mathbb{K}$ is not a critical point of P, then there exists a critical point $w\in \mathbb{K}$ of P such that

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{P\left(z\right)-P\left(w\right)}{z-w}}\right|\le S_{\mathbb{K}}|P^{\prime }\left(z\right)|.\end{array}$$

In the following result we derive a constraint for non-Archimedean Smale mean value constant, if exists.

Theorem 9.

Let $\mathbb{K}$ be a non-Archimedean valued field. The constant $S_{\mathbb{K}}$ in Problem 1, if exists, satisfies $S_{\mathbb{K}}\ge 1$.

Proof. 

Let $P\left(z\right):=(z-a)(z-b)\in \mathbb{K}\left[z\right]$. Then $P^{\prime }\left(z\right)=2z-(a+b)$ for all $z\in \mathbb{K}$. Let $z\in \mathbb{K}$ be such that $P^{\prime }\left(z\right)\ne 0$. Note that $P^{\prime }$ has only one zero, namely $c:={\displaystyle \frac{a+b}{2}}$. Define $x:=z-a$ and $y:=z-b$. Then

$$\begin{array}{cc}\hfill & z-c=z-{\displaystyle \frac{a+b}{2}}={\displaystyle \frac{z-a+z-b}{2}}={\displaystyle \frac{x+y}{2}},\hfill \\ \hfill & P^{\prime }\left(z\right)=z-a+z-b=x+y,\hfill \\ \hfill & P\left(c\right)=\left({\displaystyle \frac{a+b}{2}}-a\right)\left({\displaystyle \frac{a+b}{2}}-b\right)={\displaystyle \frac{-{(b-a)}^2}{4}}={\displaystyle \frac{-{(x-y)}^2}{4}}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill P\left(z\right)-P\left(c\right)=xy+{\displaystyle \frac{{(x-y)}^2}{4}}={\displaystyle \frac{{(x+y)}^2}{4}}.\end{array}$$

Hence

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{P\left(z\right)-P\left(c\right)}{z-c}}\right|=\left|{\displaystyle \frac{x+y}{2}}\right|={\displaystyle \frac{1}{|2|}}|P^{\prime }\left(z\right)|.\end{array}$$

Therefore

$$\begin{array}{c}\hfill S_{\mathbb{K}}\ge {\displaystyle \frac{1}{|2|}}\ge 1.\end{array}$$

□

Theorem 10.

Let $\mathbb{K}$ be a non-Archimedean valued field. Assume that there is an $n\in \mathbb{N}\setminus \{1,2\}$ such that $\mathbb{K}$ has $(n-1)$-th roots of unity. The constant $S_{\mathbb{K}}$ in Problem 1, if exists, satisfies $S_{\mathbb{K}}\ge {\displaystyle \frac{|n-1|}{|n|}}=\left|1-{\displaystyle \frac{1}{n}}\right|$.

Proof. 

We define the polynomial $P\left(z\right):=z^n-nz\in \mathbb{K}\left[z\right]$. The point $z=0$ is not a critical point of P. Let $w\in \mathbb{K}$ be any critical point of P. Then w is a $(n-1)$-th root of unity. We now see that

$$\begin{array}{cc}\hfill {\displaystyle \frac{|P(0)-P(w)|}{|0-w|}}& ={\displaystyle \frac{|P(w)|}{|w|}}=|w^n-nw|=|w^{n-1}-n|\hfill \\ \hfill & =|1-n|={\displaystyle \frac{|1-n|}{|n|}}|n|={\displaystyle \frac{|1-n|}{|n|}}|P^{\prime }\left(0\right)|.\hfill \end{array}$$

Hence $S_{\mathbb{K}}\ge {\displaystyle \frac{|n-1|}{|n|}}=\left|1-{\displaystyle \frac{1}{n}}\right|$. □

Based on Theorem 5 we formulate the following higher order problem.

Problem 2.

(Higher Order Non-Archimedean Smale Mean Value Problem) Let $\mathbb{K}$ be a non-Archimedean (complete) valued field. Whether there is a universal constant $R_{\mathbb{K}}$ (which may depend upon $\mathbb{K}$) satisfying the following: Let $P\left(z\right):=a_0+a_{1}z+a_2{z}^2+\cdots +a_n{z}^n$ be a polynomial of degree $n\ge 2$ over $\mathbb{K}$. If $z\in \mathbb{K}$ is not a critical point of P, then there exists a critical point $w\in \mathbb{K}$ of P such that

$$\begin{array}{c}\hfill {\displaystyle \frac{|P^{\left(k\right)}\left(z\right)|}{k!}}{\displaystyle \frac{{|P\left(z\right)-P\left(w\right)|}^{k-1}}{|P^{\prime }{\left(z\right)|}^k}}\le R_{\mathbb{K}}^{k-1},\forall 2\le k\le n.\end{array}$$

Theorem 11.

Let $\mathbb{K}$ be a non-Archimedean valued field. The constant $R_{\mathbb{K}}$ in Problem 2, if exists, satisfies $R_{\mathbb{K}}\ge 1/(2|2|)$.

Proof. 

We continue from the proof of Theorem 9. Let $z\in \mathbb{K}$ be not a zero of $P^{\prime }$. We have $P^{\prime \prime }\left(z\right)=2$. Hence

$$\begin{array}{c}\hfill |P^{\prime \prime }\left(z\right)|=|2|,|P^{\prime }\left(z\right)|=|x+y|,|P\left(z\right)-P\left(c\right)|={\displaystyle \frac{{|x+y|}^2}{|4|}}.\end{array}$$

We find

$$\begin{array}{c}\hfill {\displaystyle \frac{|P^{\left(2\right)}\left(z\right)|}{2!}}{\displaystyle \frac{{|P\left(z\right)-P\left(c\right)|}^{2-1}}{|P^{\prime }{\left(z\right)|}^2}}={\displaystyle \frac{|2|}{2}}{\displaystyle \frac{\left(\frac{{|x+y|}^2}{|4|}\right)}{{|x+y|}^2}}={\displaystyle \frac{1}{2|2|}}.\end{array}$$

Therefore

$$\begin{array}{c}\hfill R_{\mathbb{K}}\ge {\displaystyle \frac{1}{2|2|}}.\end{array}$$

□

3. Non-Archimedean Dubinin-Sugawa Dual Mean Value Problem

We ask non-Archimedean version of Theorem 6 as follows.

Problem 3.

(Non-Archimedean Dubinin-Sugawa Dual Mean Value Problem) Let $\mathbb{K}$ be a non-Archimedean (complete) valued field. Whether there is a universal constant $D_{\mathbb{K}}$ (which may depend upon $\mathbb{K}$) satisfying the following: Let $P\left(z\right):=a_0+a_{1}z+a_2{z}^2+\cdots +a_n{z}^n$ be a polynomial of degree $n\ge 2$ over $\mathbb{K}$. If $z\in \mathbb{K}$ is not a critical point of P, then there exists a critical point $w\in \mathbb{K}$ of P such that

$$\begin{array}{c}\hfill D_{\mathbb{K}}|P^{\prime }\left(z\right)|\le {\displaystyle \frac{|P(z)-P(w)|}{|z-w|}}.\end{array}$$

Theorem 12.

Let $\mathbb{K}$ be a non-Archimedean valued field. The constant $D_{\mathbb{K}}$ in Problem 3, if exists, satisfies $D_{\mathbb{K}}\le 1$.

Proof. 

From the proof of Theorem 9, we get that $D_{\mathbb{K}}\le |2|\le 1$. □

Theorem 13.

Let $\mathbb{K}$ be a non-Archimedean valued field. Assume that there is an $n\in \mathbb{N}\setminus \{1,2\}$ such that $\mathbb{K}$ has $(n-1)$-th roots of unity. The constant $D_{\mathbb{K}}$ in Problem 3, if exists, satisfies $D_{\mathbb{K}}\le {\displaystyle \frac{|n|}{|n-1|}}$.

Proof. 

By defining polynomial $P\left(z\right):=z^n-nz\in \mathbb{K}\left[z\right]$ we get that $D_{\mathbb{K}}\le {\displaystyle \frac{|n|}{|n-1|}}$. □

4. Conclusion

For 45 years, the Smale mean value theorem has remained a cornerstone result in the theory of polynomials, while the Dubinin–Sugawa dual mean value theorem has served a similar vital role for 17 years. In this article, we asked for non-Archimedean versions of both theorems. We established explicit bounds for these constants, conditioning on their existence.