011 Grigor Barsegian

Grigor Barsegian (National Academy of Sciences of Armenia) // A new property of arbitrary complex polynomials // For simplicity we consider an arbitrary complex monic polynomial, i.e. $P(z)=z^{m}+b_{1}z^{m-1}...+b_{m}$. Denote by $Z(a)$ the set $z_{i}(a)$ of $a$-points of $P(z)$, i.e. points $ z_{i}(a)$, $=1,2,...m$, where $P(z_{i}(a))=a$.\medskip \noindent \textbf{Theorem 1}. \textit{For an arbitrary monic polynomial }$ P(z)$\textit{\ and an arbitrary different }$a_{1},a_{2},a_{3}\in \mathbb{C}$ \textit{\ there is a point }$z^{\ast }\in Z(a_{1})\cup Z(a_{2})\cup Z(a_{3})$ \textit{\ such that} \begin{equation} |P^{\prime }(z^{\ast })|>C^{\ast }\sqrt{m}, \tag{1} \end{equation}% \textit{where }$C^{\ast }$\textit{\ is a constant depending only on }$ a_{1},a_{2},a_{3}$.\medskip \noindent \textbf{Sharpness}. The inequality (1) is sharp up to the constant.