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

TL;DR

This paper develops a compact, sharp generalization of Bernstein-type inequalities for polynomials preserved under the modified Smirnov operator $ ilde{oldsymbol{S}}_a$. For a polynomial $P$ of degree $n$ and parameters $|oldsymbol{ ilde{S}}_a[P]|$, $|eta| obreak o 1$, with $R obreak\ge 1$, the authors prove the bound

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 $α, β$ satisfying $|α| \leq 1$, $|β| \leq 1$, 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.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Corollary 2.1
• Corollary 2.2
• Theorem 2.2