On an operator preserving inequalities between polynomials

Abstract

Let $P(z)$ be a polynomial of degree n. We consider an operator $D\alpha$ that maps $P(z)$ into $D\alpha P(z) := nP(z)+(\alpha z)P\prime (z)$ and establish some results concerning the estimates of $| D\alpha P(z)|$ on the disk $| z| = R \geq 1$, and thereby obtain extensions and generalizations of a number of well-known polynomial inequalities.