Some Compact Generalization of Bernstein-Type Inequalities Preserved by Modified Smirnov Operator

Abstract

Let $P(z)$ be a polynomial of degree $n$. In $2004$, Aziz and Rather \cite{aziz2004some} investigated the dependence of \[\bigg|P(Rz)-αP(z)+β\biggl\{\biggl(\frac{R+1}{2}\biggr)^n-|α|\biggr\}P(z)\bigg|, \ \text{for} \ z \in B(\mathbb{D}),\] on ${\max }_{z\in B(\mathbb{D})}|P(z)|$, for every real and complex number $\hat{I}\pm ,{\hat{I}}^2$ satisfying $|\hat{I}\pm |\le 1$, $|{\hat{I}}^2|\le 1$, and $R\ge 1$. This paper presents a compact generalization of several well-known polynomial inequalities using modified Smirnov operator, demonstrating that the operator preserves inequalities between polynomials.

10 pages