An operator product inequalities for polynomials

TL;DR

This paper introduces a new operator for polynomials and establishes inequalities related to the product of polynomials, generalizing existing results in polynomial inequalities.

Contribution

It defines a novel operator involving derivatives of polynomials and derives generalized inequalities under specific zero location conditions.

Findings

Established a new operator for polynomials involving derivatives.

Proved generalized inequalities for polynomial products.

Extended classical results to broader polynomial classes.

Abstract

Let $P(z)$ be a polynomial of degree $n\geq 1$. In this paper we define an operator $B$, as following, $$B[P(z)]:=\lambda_0 P(z)+\lambda_1 (\frac{nz}{2}) \frac{P'(z)}{1!}+\lambda_2 (\frac{nz}{2})^2 \frac{P''(z)}{2!},$$ where $\lambda_0,\lambda_1$ and $\lambda_2$ are such that all the zeros of $$u(z)=\lambda_0 +c(n,1)\lambda_1 z+c(n,2) \lambda_2 z^2$$ lie in half plane $$|z|\leq |z-\frac{n}{2}|$$ and obtain a new generalization of some well-known results.