On the Finite-Increment Theorem for Complex Polynomials

For an arbitrary polynomial $P$ of degree at most $n$ and any points $z_1$ and $z_2$ on the complex plane, we establish estimates of the form
$$|P(z_1)-P(z_2)|\ge d_n|P'(z_1)||z_1-\zeta|,$$
where $\zeta$ is one of the roots of the equation $P(z)=P(z_2)$, and $d_n$ is a positive constant depending only on the number $n$.