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.
Citation Example:Mir A. On an operator preserving inequalities between polynomials // Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1061-1072.