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 \[\bigg|P(Rz)-\alpha P(z)+\beta\biggl\{\biggl(\frac{R+1}{2}\biggr)^n-|\alpha|\biggr\}P(z)\bigg|, \ \text{for} \ z \in B(\mathbb{D}),\] on $\max_{z\in B(\mathbb{D})}|P(z)|$, for every real and complex number $\alpha, \beta$ satisfying $|\alpha| \leq 1$, $|\beta| \leq 1$, and $R \geq 1$. This paper presents a compact generalization of several well-known polynomial inequalities using modified Smirnov operator, demonstrating that the operator preserves inequalities between polynomials.