%\documentclass[12pt]{article}
\documentclass{amsart}
%	options include 12pt or 11pt or 10pt
%	classes include article, report, book, letter, thesis
\usepackage{setspace}
\usepackage{a4}
\usepackage{amssymb,amsmath,amsthm,latexsym}
\usepackage{amsfonts}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{textcomp}
\usepackage{cite}
\usepackage{enumerate}
\usepackage[mathscr]{euscript}
\usepackage{mathtools}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem] {Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{lemma} [theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{question}[theorem] {Question}
\setlength{\parindent}{0pt} \setlength{\evensidemargin}{0.3cm}
\setlength{\oddsidemargin}{0.3cm} \setlength{\topmargin}{-1cm}
\textwidth 16cm \textheight 23cm
\onehalfspacing
\title{This is the title}
\usepackage{hyperref}
\usepackage{enumerate}
\usepackage{mathtools}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{textcomp}
\usetikzlibrary{cd}
\newcommand{\Cross}{\mathbin{\tikz [x=1.4ex,y=1.4ex,line width=.2ex] \draw (0,0) -- (1,1) (0,1) -- (1,0);}}%
\raggedbottom

\newcommand{\abs}[1]{\lvert#1\rvert}


\newcommand{\blankbox}[2]{%
	\parbox{\columnwidth}{\centering
	
		\setlength{\fboxsep}{0pt}%
		\fbox{\raisebox{0pt}[#2]{\hspace{#1}}}%
	}%
}
\usepackage{fancyhdr}


\pagestyle{fancy}

\fancyhead[LO]{\textbf{Non-Archimedean Smale Mean Value and Dubinin–Sugawa Dual Mean Value Problems}}
\fancyhead[RE]{\textbf{K. MAHESH KRISHNA}}







\begin{document}
		\vspace{0.9cm}	
	\hrule\hrule\hrule\hrule\hrule
	\vspace{0.3cm}	
\begin{center}
{\bf{Non-Archimedean Smale Mean Value and Dubinin–Sugawa Dual Mean Value Problems}\\
	\vspace{.5cm}
	\hrule\hrule\hrule\hrule
		\vspace{.5cm}
K. Mahesh Krishna}  \\
School of Mathematics and Natural Sciences\\
Chanakya University Global Campus\\
NH-648, Haraluru Village\\
Devanahalli Taluk, 	Bengaluru  North District\\
Karnataka State, 562 110, India \\
Email: kmaheshak@gmail.com\\


Date: \today
\end{center}

\hrule\hrule
\vspace{0.5cm}
%--------------------------------------
\textbf{Abstract}: Motivated by the Smale mean value and the Dubinin–Sugawa dual mean value theorems, we formulate non-Archimedean Smale mean value and Dubinin–Sugawa dual mean value problems. We show that if a universal constant satisfying the non-Archimedean Smale mean value problem exists, it must be greater than or equal to one and if a universal constant satisfying the non-Archimedean Dubinin–Sugawa dual mean value problem exists, it must be less than or equal to one. We also derive other bounds whenever roots of unity exist in a non-Archimedean field.



\textbf{Keywords}: Smale mean value theorem, Dubinin-Sugawa dual mean value theorem, Non-Archimedean field.


\textbf{Mathematics Subject Classification (2020)}: 30C10, 11C08, 11S05.\\


\hrule
%\tableofcontents
\hrule
\section{Introduction}
In 1981, Steve Smale proved the following result using the Koebe 1/4-theorem \cite{SMALE1981}.
\begin{theorem}\cite{SMALE1981} \textbf{(Smale Mean Value Theorem)} \label{SMALEMEANTHEOREM}
	Let $P(z)\coloneqq a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\geq 2$ over $\mathbb{C}$. If $z\in\mathbb{C}$ is not a critical point of $P$ (i.e., $P'(z)\neq 0$), then 
	\begin{align}\label{SMALEIN}
		\min _{w\in \mathbb{C}, P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|\leq 4 |P'(z)|.
	\end{align}
\end{theorem}
Note that Theorem \ref{SMALEMEANTHEOREM} can also be stated as follows. 
\begin{theorem} \cite{SMALE1981} \textbf{(Smale Mean Value Theorem)}\label{SMALETHEOREM}
Let $P(z)\coloneqq a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\geq 2$ 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{align}\label{CRIT}
	\left|\frac{P(z)-P(w)}{z-w}\right|\leq 4 |P'(z)|.
\end{align}	
\end{theorem}
The proof of Theorem \ref{SMALETHEOREM} can also be found in books \cite{RAHMANSCHMEISSERBOOK, SHEILBOOK}. Inequality (\ref{SMALEIN}) led Smale to ask the following problem. What is the best constant $4>c>0$ such that the Inequality (\ref{SMALEIN})  can be replaced by 
\begin{align}\label{SMALEC}
	\min _{w\in \mathbb{C}, P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|\leq c |P'(z)|?	
\end{align}
Smale made the following conjecture with regard to Inequality (\ref{SMALEC}), which remains  open today.
\begin{conjecture} \cite{SMALEIN, SMALE1981, SMALE2000, SHUBSMALE2, SMALE1985, SHUBSMALE1}\label{SMALECONJECTURE} \textbf{(Smale Mean Value Conjecture)}
	Let $P(z)\coloneqq a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\geq 2$ over $\mathbb{C}$. If $z\in\mathbb{C}$ is not a critical point of $P$, then 
\begin{align*}
	\min _{w\in \mathbb{C}, P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|\leq 1 |P'(z)|
\end{align*}	
or 
\begin{align}\label{1}
	\min _{w\in \mathbb{C}, P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|\leq \frac{n-1}{n} |P'(z)|=\frac{\operatorname{deg} (P)-1}{\operatorname{deg} (P)} |P'(z)|.
\end{align}
\end{conjecture}
By taking polynomial $P(z)\coloneqq z^n-nz \in \mathbb{C}[z]$, we see that the constant $(n-1)/n$ in inequality (\ref{1}) cannot be improved further. Conjecture  \ref{SMALECONJECTURE} also appears as  Problem 1 in the Addenda after the list of 18 open problems of Smale in \textbf{Mathematical Problems for the  Next Century} \cite{SMALE2000}. Now for a polynomial $P$  of degree $n\geq2$, define 
\begin{align*}
	S(P)\coloneqq \sup_{z\in \mathbb{C},P'(z)\neq 0}\frac{1}{|P'(z)|}	\min _{w\in \mathbb{C}, P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|
\end{align*}
and 
\begin{align*}
	M\coloneqq \sup_{P \text{ is a non linear polynomial}} S(P).
\end{align*}
 Smale conjecture then asks whether $M=1$  or $S(P)=\frac{\text{deg}(P)-1}{\text{deg}(P)}$? It is known that the Conjecture \ref{SMALECONJECTURE} holds for polynomials of degree 2, 3, 4, \cite{SHEILBOOK, TISCHLER, RAHMANSCHMEISSERBOOK}. Through computer calculations, Smale conjecture is known to hold for polynomials of degree upto 10 \cite{SENDOVMARINOV}. 

Usually,  the study of Smale mean value conjecture goes with the following conjecture which is known as normalization equivalence to Conjecture  \ref{SMALECONJECTURE}.
\begin{conjecture}\cite{SMALE1981,SHEILBOOK, RAHMANSCHMEISSERBOOK}\label{SMALENORMAL}
	 \textbf{(Normalized Smale Mean Value Conjecture)} Let $P$ be a degree $n\geq2$ polynomial  over $\mathbb{C}$ which is normalized, i.e.,  $P(0)=0$ and $P'(0)=1$. Then 
	\begin{align*}
		S_0(P)\coloneqq 	\min _{w\in \mathbb{C}, P'(w)=0}\left|\frac{P(w)}{w}\right|=\frac{\operatorname{deg}(P)-1}{\operatorname{deg}(P)}
	\end{align*}
	or 
	\begin{align*}
		M_0\coloneqq \sup_{P \operatorname{ is \, a \, non \, linear \, polynomial}, P(0)=0, P'(0)=1} S_0(P)=1.	
	\end{align*}
\end{conjecture}
In 1996, Andrievskii  and Ruscheweyh  showed that  if  critical points of a normalized polynomial $P$ lie on the unit circle, then $S_0(P)\leq 1$ \cite{ANDRRUS}.  In 2009, Hinkkanen and  Kayumov showed that if all the critical points of a normalized polynomial $P$ are real, then $S_0(P)\leq \frac{2}{3}$ \cite{HINKKANENKAYUMOV2}. 
  In 2010, Hinkkanen  and Kayumov proved that if the critical points of normalized polynomial $P$ lie on the sector $\{re^{i\theta}:r>0,|\theta|\leq \pi/6\}$, then $S_0(P)\leq \frac{1}{2}$   and if the critical points of normalized polynomial  $P$ lie in the union of two rays $\{1+re^{\pm i\theta}: r\geq 0\}$ for each $0<\theta\leq \pi/2$, then $S_0(P)\leq \frac{2}{3}$  \cite{HINKKANENKAYUMOV3}. In \cite{RAHMANSCHMEISSERBOOK} it is showed that if the critical points of a normalized polynomial are all real, then $S_0(P)\leq \frac{\text{deg}(P)-2}{\text{deg}(P)}\left[\left(\frac{\text{deg}(P)-1}{\text{deg}(P)-2}\right)^{{\text{deg}(P)-1}}-2\right]$ and in  \cite{SHEILBOOK} it is showed that $S_0(P)\leq e-2$. \\
   In  2002, Beardon, Minda and Ng showed that $S(P)\leq  4^\frac{\text{deg}(P)-2}{\text{deg}(P)-1}$ \cite{BEARDONMINDANG}.  In 2006, Fijikawa and Sugawa proved that $S(P)\leq  4 \frac{1+(\text{deg}(P)-2)4^\frac{-1}{\text{deg}(P)-1}}{\text{deg}(P)+1}$ \cite{FUJIKAWASUGAWA}.  In 2007, Conte, Fujikawa and Lakic showed that $S(P)\leq  4 \frac{\text{deg}(P)-1}{\text{deg}(P)+1}$ \cite{CONTEFUJIKAWALAKIC}. In 2007, Crane showed that there exists $n_0\geq2$ such that for all $n\geq n_0$, $S(P)\leq 4-\frac{2}{\sqrt{\text{deg}(P)}}$ \cite{CRANE} (also see \cite{CRANE2006}). In 2003, Ng proved that Conjecture \ref{SMALECONJECTURE} holds  with $M=2$ for   all odd polynomials with nonzero linear terms \cite{NG}. In 2006, Dubinin proved that if the critical points of a normalized polynomial $P$ have equal modulus, then $S_0(P)\leq \frac{\operatorname{deg}(P)-1}{\operatorname{deg}(P)}$ \cite{DUBININ}.
   
   
At the other side, in 2002,  Sendov formulated a generalization of Conjecture \ref{SMALECONJECTURE}  \cite{SENDOV2002}.  In 2003, Sendov and Nikolov gave some variations of Conjecture \ref{SMALECONJECTURE} \cite{SENDOVNIKOLOV}. In 2005, Tyson showed  that Tischler's strong form of Smale mean value conjecture \cite{TISCHLER, TISCHLER1994}  fails \cite{TYSON}.  One can  interpret Conjecture \ref{SMALECONJECTURE} in the form of finite Blaschke products \cite{NGTSANGCHAPETR, NGZHANG, SHEILBOOK}. It is also possible to connect Conjecture \ref{SMALECONJECTURE} with electrostatics \cite{DIMITROV}. \\
In  1981,  Smale also derived the following higher order mean value theorem \cite{SMALE1981}. 
\begin{theorem} \cite{SMALE1981} \label{HIGHER}
\textbf{(Higher Order Smale Mean Value Theorem)} Let $P(z)\coloneqq a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\geq 2$ 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{align}\label{CRITHIGHER}
	\frac{|P^{(k)}(z)|}{k!}	\frac{|P(z)-P(w)|^{k-1}}{|P'(z)|^k}\leq 4^{k-1}, \quad \forall 2\leq k \leq n.
\end{align}	
\end{theorem}
In 2009,  Dubinin and Sugawa derived the following striking result which is a dual result for Smale mean value theorem \cite{DUBININSUGAWA}. 
\begin{theorem} \cite{DUBININSUGAWA, DUBININ2012} \textbf{(Dubinin-Sugawa Dual Mean Value Theorem)}\label{DUBINSUGAWA}
Let $P(z)\coloneqq a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\geq 2$ 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{align*}
\frac{|P'(z)|}{n4^n}	\leq \left|\frac{P(z)-P(w)}{z-w}\right|.
\end{align*}
In other words, 
\begin{align*}
\frac{|P'(z)|}{n4^n}\leq 	\max _{w\in \mathbb{C}, P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|.
\end{align*}	
\end{theorem}
Dubinin and Sugawa made  the following conjecture in 2009 \cite{DUBININSUGAWA}.
\begin{conjecture}\cite{DUBININSUGAWA, DUBININ2012} \label{DUBINSUGAWACONJECTURE} \textbf{(Dubinin-Sugawa Dual Mean Value Conjecture)}
	Let $P(z)\coloneqq a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\geq 2$ over $\mathbb{C}$. If $z\in\mathbb{C}$ is not a critical point of $P$, then 
\begin{align*}
\frac{|P'(z)|}{n}	\leq \max _{w\in \mathbb{C}, P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|.
\end{align*}
\end{conjecture}
For a polynomial $P$  of degree $n\geq2$, define 
\begin{align*}
	DS(P)\coloneqq \inf_{z\in \mathbb{C},P'(z)\neq 0}\frac{1}{|P'(z)|}	\max _{w\in \mathbb{C}, P'(w)=0}\left|\frac{P(z)-P(w)}{z-w}\right|.
\end{align*}
%and 
%\begin{align*}
%	N\coloneqq \inf_{P \text{ is a non linear polynomial}} DS(P).
%\end{align*}
Similar to Conjecture \ref{SMALENORMAL}, we have the following normalized version of  Conjecture \ref{DUBINSUGAWACONJECTURE}.
\begin{conjecture}\cite{DUBININSUGAWA, DUBININ2012}
 \textbf{(Normalized Smale Dubinin-Sugawa Dual Mean Value Conjecture)} Let $P$ be a degree $n\geq2$ polynomial  over $\mathbb{C}$ which is normalized, i.e.,  $P(0)=0$ and $P'(0)=1$. Then 
\begin{align*}
	DS_0(P)\coloneqq 	\max _{w\in \mathbb{C}, P'(w)=0}\left|\frac{P(w)}{w}\right|=\frac{1}{n}=\frac{1}{\operatorname{deg}(P)}.
\end{align*}
\end{conjecture}
In 2016, Ng and Zhang showed that  if $P$ is a normalized polynomial, then $	DS_0(P)> \frac{1}{4^\text{deg (P)}}$ \cite{NGZHANG}. In 2019 Dubinin showed  that $	DS(P)\geq \frac{1}{n} \tan \left(\frac{\pi}{4 \text{deg (P)}}\right)$ \cite{DUBININ2019}. 
In 2019,  Hinkkanen,  Kayumov  and Khammatova showed that Conjecture \ref{DUBINSUGAWACONJECTURE} holds for polynomials of degree six \cite{HINKKANENKAYUMOVKHAMMATOVA}. In 2025,  Hinkkanen,  Kayumov  and Khammatova showed that Conjecture \ref{DUBINSUGAWACONJECTURE} holds for polynomials of degree six \cite{HINKKANENKAYUMOVKHAMMATOVA2}. 

In this paper, we ask for  non-Archimedean versions of Theorems \ref{SMALEMEANTHEOREM} and \ref{DUBINSUGAWA}. 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.




  \section{Non-Archimedean Smale Mean Value  Problem}
  Let $\mathbb{K}$ be a field. Recall that a map $|\cdot|:\mathbb{K} \to [0, \infty)$ is said to be a \textbf{non-Archimedean valuation} if the following conditions hold:
  \begin{enumerate}[\upshape(i)]
  	\item If $\lambda \in \mathbb{K}$ satisfies $|\lambda|=0$, then $\lambda=0$. 
  	\item $|\lambda\mu|=|\lambda||\mu|$ for all $\lambda, \mu \in \mathbb{K}$.
  	\item (Ultratriangle inequality) $|\lambda+\mu|\leq \max\{\lambda|, |\mu|\}$ for all $\lambda, \mu \in \mathbb{K}$.
  \end{enumerate}
  We say $\mathbb{K}$ is non-Archimedean complete valued field if it is complete w.r.t. the ultrametric $d(\lambda, \mu)\coloneqq |\lambda-\mu|$ for all $\lambda, \mu \in \mathbb{K}$ \cite{SCHIKHOF}. Throughout the article we assume that $\mathbb{K}$ is algebraically closed.

  
  We ask for non-Archimedean version of Theorem \ref{SMALETHEOREM} as follows.
  \begin{problem} \textbf{(Non-Archimedean Smale Mean Value  Problem)}\label{CSMALE}
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 $P(z)\coloneqq a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\geq 2$ 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{align*}
	\left|\frac{P(z)-P(w)}{z-w}\right|\leq S_\mathbb{K} |P'(z)|.
\end{align*}	
  \end{problem}
In the following result we derive a constraint for non-Archimedean Smale mean value constant, if exists.
\begin{theorem}\label{CH}
Let $\mathbb{K}$ be a non-Archimedean valued field. The constant $S_\mathbb{K}$  in Problem \ref{CSMALE}, if exists, satisfies   $S_\mathbb{K}\geq1$. 
  \end{theorem}
  \begin{proof}
 Let  $P(z)\coloneqq (z-a)(z-b)  \in \mathbb{K}[z]$. Then $P'(z)=2z-(a+b)$ for all $z\in \mathbb{K}$. Let $z\in \mathbb{K}$ be such that $P'(z)\neq0$. Note that $P'$ has only one zero, namely $c\coloneqq\frac{a+b}{2}$. 
Define $x\coloneqq z-a$ and $y\coloneqq z-b$.   Then
  \begin{align*}
  	&z-c=z-\frac{a+b}{2}=\frac{z-a+z-b}{2}=\frac{x+y}{2},\\
  	&P'(z)=z-a+z-b=x+y,\\
  	&P(c)=\left(\frac{a+b}{2}-a\right)\left(\frac{a+b}{2}-b\right)=\frac{-(b-a)^2}{4}=\frac{-(x-y)^2}{4}
  \end{align*}
and 
\begin{align*}
P(z)-P(c)=xy+\frac{(x-y)^2}{4}=\frac{(x+y)^2}{4}.
\end{align*}
Hence
\begin{align*}
	\left|\frac{P(z)-P(c)}{z-c}\right|=\left|\frac{x+y}{2}\right|= \frac{1}{|2|}|P'(z)|.
\end{align*}
Therefore
\begin{align*}
	S_\mathbb{K}\geq \frac{1}{|2|}\geq 1.
\end{align*}
  \end{proof}
\begin{theorem}
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 \ref{CSMALE}, if exists, satisfies   $S_\mathbb{K}\geq \frac{|n-1|}{|n|}=\left|1-\frac{1}{n}\right|$. 	
\end{theorem}
\begin{proof}
	We define the polynomial  $P(z)\coloneqq z^{n}-nz \in \mathbb{K}[z]$. 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{align*}
	\frac{|P(0)-P(w)|}{|0-w|}&=\frac{|P(w)|}{|w|}=|w^n-nw|=|w^{n-1}-n|\\
	&=|1-n|=\frac{|1-n|}{|n|}|n|=\frac{|1-n|}{|n|}|P'(0)|.
\end{align*} 
	Hence $S_\mathbb{K}\geq \frac{|n-1|}{|n|}=\left|1-\frac{1}{n}\right|$.
\end{proof}
Based on Theorem \ref{HIGHER} we formulate the following higher order problem. 
\begin{problem}\label{HIGHERMEAN}
\textbf{(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 $P(z)\coloneqq a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\geq 2$ 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{align*}
	\frac{|P^{(k)}(z)|}{k!}	\frac{|P(z)-P(w)|^{k-1}}{|P'(z)|^k}\leq R_\mathbb{K}^{k-1}, \quad \forall 2\leq k \leq n.
\end{align*}	
\end{problem}
  \begin{theorem}
Let $\mathbb{K}$ be a non-Archimedean valued field. The constant $R_\mathbb{K}$  in Problem \ref{HIGHERMEAN}, if exists, satisfies   $R_\mathbb{K}\geq1/(2|2|)$. 	
\end{theorem}
\begin{proof}
We continue from the proof of Theorem \ref{CH}. Let $z\in \mathbb{K}$ be not a zero of $P'$. We have $P''(z)=2$. Hence 
\begin{align*}
	|P''(z)|=|2|, \quad |P'(z)|=|x+y|,\quad 
	|P(z)-P(c)|=\frac{|x+y|^2}{|4|}.
\end{align*}
We find
\begin{align*}
\frac{|P^{(2)}(z)|}{2!}	\frac{|P(z)-P(c)|^{2-1}}{|P'(z)|^2}=\frac{|2|}{2}	\frac{\left(\frac{|x+y|^2}{|4|}\right)}{|x+y|^2}=\frac{1}{2|2|}.
\end{align*}
Therefore 
\begin{align*}
	R_\mathbb{K}\geq \frac{1}{2|2|}.
\end{align*}
\end{proof}   




  
 \section{Non-Archimedean Dubinin-Sugawa Dual  Mean Value  Problem}
 We ask non-Archimedean version of Theorem \ref{DUBINSUGAWA} as follows.
 \begin{problem}\textbf{(Non-Archimedean Dubinin-Sugawa Dual  Mean Value  Problem)}\label{DUALSMALE}
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 $P(z)\coloneqq a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a polynomial of degree $n\geq 2$ 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{align*}
 D_\mathbb{K}|P'(z)|	\leq  \frac{|P(z)-P(w)|}{|z-w|}.
\end{align*}
 \end{problem}

 \begin{theorem}
 Let $\mathbb{K}$ be a non-Archimedean valued field. The constant $D_\mathbb{K}$  in Problem \ref{DUALSMALE}, if exists, satisfies   $D_\mathbb{K}\leq1$. 	
 \end{theorem}
 \begin{proof}
  From the proof of Theorem \ref{CH}, we get that $D_\mathbb{K}\leq |2|\leq 1$.  
 \end{proof}
 \begin{theorem}
 	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 \ref{DUALSMALE}, if exists, satisfies   $D_\mathbb{K}\leq \frac{|n|}{|n-1|}$. 	
 \end{theorem}
 \begin{proof}
 	By defining polynomial  $P(z)\coloneqq z^{n}-nz \in \mathbb{K}[z]$ we get that $D_\mathbb{K}\leq \frac{|n|}{|n-1|}$.
 \end{proof}


\section{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.





 \bibliographystyle{plain}
\begin{thebibliography}{10}
	
	\bibitem{ANDRRUS}
	Vladimir~V. Andrievskii and Stephan Ruscheweyh.
	\newblock {\em Complex polynomials and maximal ranges: background and
		applications}, volume 430 of {\em Math. Appl.}
	\newblock Kluwer Acad. Publ., Dordrecht, 1998.
	
	\bibitem{BEARDONMINDANG}
	A.~F. Beardon, D.~Minda, and T.~W. Ng.
	\newblock Smale's mean value conjecture and the hyperbolic metric.
	\newblock {\em Math. Ann.}, 322(4):623--632, 2002.
	
	\bibitem{CONTEFUJIKAWALAKIC}
	Anthony Conte, Ege Fujikawa, and Nikola Lakic.
	\newblock Smale's mean value conjecture and the coefficients of univalent
	functions.
	\newblock {\em Proc. Amer. Math. Soc.}, 135(10):3295--3300, 2007.
	
	\bibitem{CRANE2006}
	Edward Crane.
	\newblock Extremal polynomials in {S}male's mean value conjecture.
	\newblock {\em Comput. Methods Funct. Theory}, 6(1):145--163, 2006.
	
	\bibitem{CRANE}
	Edward Crane.
	\newblock A bound for {S}male's mean value conjecture for complex polynomials.
	\newblock {\em Bull. Lond. Math. Soc.}, 39(5):781--791, 2007.
	
	\bibitem{DIMITROV}
	Dimitar~K. Dimitrov.
	\newblock Smale's conjecture on mean values of polynomials and electrostatics.
	\newblock {\em Serdica Math. J.}, 33(4):399--410, 2007.
	
	\bibitem{DUBININ2012}
	V.~N. Dubinin.
	\newblock Methods of geometric function theory in classical and modern problems
	for polynomials.
	\newblock {\em Uspekhi Mat. Nauk}, 67(4(406)):3--88, 2012.
	
	\bibitem{DUBININ2019}
	V.~N. Dubinin.
	\newblock On the dual mean-value conjecture for complex polynomials.
	\newblock {\em Math. Notes}, 106:133--135, 2019.
	
	\bibitem{DUBININSUGAWA}
	Vladimir Dubinin and Toshiyuki Sugawa.
	\newblock Dual mean value problem for complex polynomials.
	\newblock {\em Proc. Japan Acad. Ser. A Math. Sci.}, 85(9):135--137, 2009.
	
	\bibitem{DUBININ}
	V.N. Dubinin.
	\newblock Inequalities for critical values of polynomials.
	\newblock {\em Sbornik: Mathematics}, 197(8):1167--1176, 2006.
	
	\bibitem{FUJIKAWASUGAWA}
	Ege Fujikawa and Toshiyuki Sugawa.
	\newblock Geometric function theory and {S}male's mean value conjecture.
	\newblock {\em Proc. Japan Acad. Ser. A Math. Sci.}, 82(7):97--100, 2006.
	
	\bibitem{HINKKANENKAYUMOV2}
	Aimo Hinkkanen and Ilgiz Kayumov.
	\newblock On critical values of polynomials with real critical points.
	\newblock {\em Constr. Approx.}, 32(2):385--392, 2010.
	
	\bibitem{HINKKANENKAYUMOV3}
	Aimo Hinkkanen and Ilgiz Kayumov.
	\newblock Smale's problem for critical points on certain two rays.
	\newblock {\em J. Aust. Math. Soc.}, 88(2):183--191, 2010.
	
	\bibitem{HINKKANENKAYUMOVKHAMMATOVA}
	Aimo Hinkkanen, Ilgiz~R. Kayumov, and Diana~M. Khammatova.
	\newblock Dual {S}male's mean value conjecture.
	\newblock {\em Proc. Amer. Math. Soc.}, 147(12):5227--5237, 2019.
	
	\bibitem{HINKKANENKAYUMOVKHAMMATOVA2}
	Aimo Hinkkanen, Ilgiz~R. Kayumov, and Diana~M. Khammatova.
	\newblock Dual {Smale}'s mean value conjecture for {{\(n=7\)}}.
	\newblock {\em Constr. Approx.}, 61(3):603--613, 2025.
	
	\bibitem{NG}
	T.~W. Ng.
	\newblock Smale's mean value conjecture for odd polynomials.
	\newblock {\em J. Aust. Math. Soc.}, 75(3):409--411, 2003.
	
	\bibitem{NGZHANG}
	Tuen-Wai Ng and Yongquan Zhang.
	\newblock Smale's mean value conjecture for finite {B}laschke products.
	\newblock {\em J. Anal.}, 24(2):331--345, 2016.
	
	\bibitem{NGTSANGCHAPETR}
	T.W. Ng and C.Y. Tsang.
	\newblock {\em Polynomials Versus Finite {B}laschke Products}, volume~65 of
	{\em Fields Institute Communications}.
	\newblock Springer, Boston, MA., 2013.
	
	\bibitem{RAHMANSCHMEISSERBOOK}
	Q.~I. Rahman and G.~Schmeisser.
	\newblock {\em Analytic theory of polynomials}, volume~26 of {\em London
		Mathematical Society Monographs. New Series}.
	\newblock The Clarendon Press, Oxford University Press, Oxford, 2002.
	
	\bibitem{SCHIKHOF}
	W.~H. Schikhof.
	\newblock {\em Ultrametric calculus. {An} introduction to {{\(p\)}}-adic
		analysis}, volume~4.
	\newblock Cambridge University Press, Cambridge, 1984.
	
	\bibitem{SENDOV2002}
	Bl. Sendov.
	\newblock Generalization of the {S}male's mean value conjecture.
	\newblock {\em C. R. Acad. Bulgare Sci.}, 55(11):5--10, 2002.
	
	\bibitem{SENDOVNIKOLOV}
	Bl. Sendov and N.~Nikolov.
	\newblock Variations of {S}male's mean value conjecture.
	\newblock {\em C. R. Acad. Bulgare Sci.}, 56(11):9--14, 2003.
	
	\bibitem{SENDOVMARINOV}
	Blagovest Sendov and Pencho Marinov.
	\newblock Verification of {S}male's mean value conjecture for {$n\leq 10$}.
	\newblock {\em C. R. Acad. Bulgare Sci.}, 60(11):1151--1156, 2007.
	
	\bibitem{SHEILBOOK}
	T.~Sheil-Small.
	\newblock {\em Complex polynomials}, volume~75 of {\em Cambridge Studies in
		Advanced Mathematics}.
	\newblock Cambridge University Press, Cambridge, 2002.
	
	\bibitem{SHUBSMALE2}
	M.~Shub and S.~Smale.
	\newblock Computational complexity: on the geometry of polynomials and a theory
	of cost. {II}.
	\newblock {\em SIAM J. Comput.}, 15(1):145--161, 1986.
	
	\bibitem{SHUBSMALE1}
	Mike Shub and Steven Smale.
	\newblock Computational complexity. {O}n the geometry of polynomials and a
	theory of cost. {I}.
	\newblock {\em Ann. Sci. \'{E}cole Norm. Sup. (4)}, 18(1):107--142, 1985.
	
	\bibitem{SMALE1981}
	Steve Smale.
	\newblock The fundamental theorem of algebra and complexity theory.
	\newblock {\em Bull. Amer. Math. Soc. (N.S.)}, 4(1):1--36, 1981.
	
	\bibitem{SMALE1985}
	Steve Smale.
	\newblock On the efficiency of algorithms of analysis.
	\newblock {\em Bull. Amer. Math. Soc. (N.S.)}, 13(2):87--121, 1985.
	
	\bibitem{SMALEIN}
	Steve Smale.
	\newblock Mathematical problems for the next century.
	\newblock {\em Math. Intelligencer}, 20(2):7--15, 1998.
	
	\bibitem{SMALE2000}
	Steve Smale.
	\newblock {\em Mathematical problems for the next century}.
	\newblock Amer. Math. Soc., Providence, RI, 2000.
	
	\bibitem{TISCHLER}
	David Tischler.
	\newblock Critical points and values of complex polynomials.
	\newblock {\em J. Complexity}, 5(4):438--456, 1989.
	
	\bibitem{TISCHLER1994}
	David Tischler.
	\newblock Perturbations of critical fixed points of analytic maps.
	\newblock {\em Ast\'{e}risque}, (222):8, 407--422, 1994.
	
	\bibitem{TYSON}
	Jeremy~T. Tyson.
	\newblock Counterexamples to {T}ischler's strong form of {S}male's mean value
	conjecture.
	\newblock {\em Bull. London Math. Soc.}, 37(1):95--106, 2005.
	
\end{thebibliography}

 


\end{document}