mean value problem

Let $d\geq 2$ be an integer, let

$$P(w)=a_d w^d+a_{d-1}w^{d-1}+\cdots+a_1 w+a_0,\qquad a_j\in\mathbb{C},\ a_d\neq 0,$$

be a polynomial of degree $d$ over $\mathbb{C}$, and call $c\in\mathbb{C}$ a critical point of $P$ if $P'(c)=0$. Let $z\in\mathbb{C}$ be a point with $P'(z)\neq 0$, so that $z$ is not itself a critical point.

Then there exists a critical point $c$ of $P$ such that

$$\left|\frac{P(z)-P(c)}{z-c}\right|\leq K\,|P'(z)|\qquad\text{with }K=1,$$

the constant $K$ being independent of $d$, of $P$, and of $z$.