A new property of arbitrary complex polynomials

A new property for an arbitrary complex polynomial is discovered, which relates to its derivative.

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 $z_{i}(a)$, $i=1,2,...m$, where $P(z_{i}(a))=a$.

Theorem 1. For an arbitrary monic polynomial $P(z)$ and an arbitrary different $a_{1},a_{2},a_{3}\in \mathbb{C}$ there is a point $z^{\ast }\in Z(a_{1})\cup Z(a_{2})\cup Z(a_{3})$ such that

$|P^{\prime }(z^{\ast })|>C^{\ast }\sqrt{m},  \tag{1}$

where $C^{\ast }$ is a constant depending only on $a_{1},a_{2},a_{3}$.

Sharpness. The inequality (1) is sharp up to the constant.