C*-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture

Abstract: Based on Smale mean value conjecture \textit{[Bull. Amer. Math. Soc., 1981]} and Dubinin-Sugawa dual mean value conjecture \textit{[Proc. Japan Acad. Ser. A Math. Sci., 2009]} we formulate the following conjectures. \textbf{C*-algebraic Smale Mean Value Conjecture : Let $\mathcal{A}$ be a commutative C*-algebra. Let $P(z)= (z-a_1)\cdots (z-a_n)$ be a polynomial of degree $n\geq 2$ over $\mathcal{A}$, $a_1, \dots, a_n \in \mathcal{A}$. If $z\in\mathcal{A}$ is not a critical point of $P$, then there exists a critical point $w\in \mathcal{A}$ of $P$ such that \begin{align*} \frac{|P(z)-P(w)|}{|z-w|}\leq 1 |P'(z)| \end{align*} or \begin{align*} \frac{|P(z)-P(w)|}{|z-w|}\leq \frac{n-1}{n} |P'(z)|=\frac{\operatorname{deg} (P)-1}{\operatorname{deg} (P)} |P'(z)|. \end{align*}} \textbf{C*-algebraic Dubinin-Sugawa Dual Mean Value Conjecture : Let $\mathcal{A}$ be a commutative C*-algebra. Let $P(z)= (z-a_1)\cdots (z-a_n)$ be a polynomial of degree $n\geq 2$ over $\mathcal{A}$, $a_1, \dots, a_n \in \mathcal{A}$. If $z\in \mathcal{A}$ is not a critical point of $P$, then there exists a critical point $w\in \mathcal{A}$ of $P$ such that \begin{align*} \frac{|P'(z)|}{\operatorname{deg} (P)} =\frac{|P'(z)|}{n} \leq \frac{|P(z)-P(w)|}{|z-w|}. \end{align*}} We show that (even a strong form of) C*-algebraic Smale mean value conjecture and C*-algebraic Dubinin-Sugawa dual mean value conjecture hold for degree 2 C*-algebraic polynomials over commutative C*-algebras.