Some inequalities of the V. A. Markov type

Let $B$ be a domain in the complex plane, let $p_n(z)$ and $P_n(z)$ be polynomials of degree $n$ where the zeros of $P_n(z)$ lie in $\overline B$, let $\varphi(z)$ be a finite function, $\varphi(z)\ne0$, $z\overline\in\overline B$. We consider the problem of estimating from above the functions $L[p_n(z)]=\varphi p_n'(z)-wp_n(z),\,\overline\in\overline B$, если $|p_n(z)|\leqslant+|P_n(z)|$ при $z\in\overline B$. Under some very general conditions on $B$, $z$, $\varphi(z)$ and $w$ we prove the inequality $|L[p_n(z)]|\leqslant|L[P_n(z)]|$.