On the zeros of certain composite polynomials and an operator preserving inequalities

TL;DR

This paper generalizes a classical zero localization result for polynomials by relaxing degree constraints and introduces a linear operator that preserves Bernstein-type inequalities, broadening the scope of polynomial inequality preservation.

Contribution

It extends Marden's zero localization theorem to polynomials of different degrees and introduces an operator that maintains Bernstein-type inequalities.

Findings

Generalized zero localization for polynomials of arbitrary degrees

Introduced a linear operator preserving Bernstein inequalities

Broadened applicability of classical polynomial zero bounds

Abstract

If all the zeros of $n$th degree polynomials $f(z)$ and $g(z) = \sum_{k=0}^{n}\lambda_k\binom{n}{k}z^k$ respectively lie in the cricular regions $|z|\leq r$ and $|z| \leq s|z-\sigma|$, $s>0$, then it was proved by Marden \cite[p. 86]{mm} that all the zeros of the polynomial $h(z)= \sum_{k=0}^{n}\lambda_k f^{(k)}(z) \frac{(\sigma z)^k}{k!}$ lie in the circle $|z| \leq r ~ \max(1,s)$. In this paper, we relax the condition that $f(z)$ and $g(z)$ are of the same degree and instead assume that $f(z)$ and $g(z)$ are polynomials of arbitrary degree $n$ and $m$ respectively, $m\leq n,$ and obtain a generalization of this result. As an application, we also introduce a linear operator which preserve Bernstein type polynomial inequalities.