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Based on Smale mean value conjecture \textit{[Bull. Amer. Math. Soc., 1981]}and Dubinin-Sugawa dual mean value conjecture \textit{[Proc. Japan Acad. Ser. AMath. Sci., 2009]} we formulate the following conjectures.  \textbf{C*-algebraic Smale Mean Value Conjecture : Let $\mathcal{A}$ be acommutative C*-algebra. Let $P(z)= (z-a_1)\cdots (z-a_n)$ be a polynomial ofdegree $n\geq 2$ 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 criticalpoint $w\in \mathcal{A}$ 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 $P(z)= (z-a_1)\cdots (z-a_n)$ bea 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 thereexists 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 formof) C*-algebraic Smale mean value conjecture and C*-algebraic Dubinin-Sugawadual mean value conjecture hold for degree 2 C*-algebraic polynomials overcommutative C*-algebras.

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