Inequalities involving a measure of Marcell\'{a}n class and zeros of corresponding orthogonal polynomials

TL;DR

This paper explores inequalities and zero distribution properties of polynomials related to the Marcellán class on the unit circle, analyzing how measure changes affect orthogonal polynomial zeros and norms.

Contribution

It introduces new inequalities and zero behavior analyses for polynomials in the Marcellán class, extending understanding of measure impacts without measure ordering.

Findings

Zeros of quasi-orthogonal and orthogonal polynomials are closely related.

New inequalities between polynomial norms involving different measures are established.

Lubinsky type inequalities are derived without measure ordering assumptions.

Abstract

Let $\tilde{\Phi}_n$ be a quasi-orthogonal polynomial of order 1 on the unit circle, obtained from an orthogonal polynomial $\Phi_n$ with measure $\mu$, which is in the Marcell\'{a}n class, if there exist another measure $\tilde{\mu}$ such that $\tilde{\Phi}_n$ is a monic orthogonal polynomial. This article aims to investigate various properties related to the Marcell\'{a}n class. At first, we study the behaviour of the zeros between $\Phi_n$ and $\tilde{\Phi}_n$. Along with numerical examples, we analyze the zeros of $\Phi_n$, its POPUC and the linear combination of the POPUC. Further, comparison of the norm inequalities among $\Phi_n$ and $\tilde{\Phi}_n$ are obtained by involving their measures. This leads to the study of the Lubinsky type inequality between the measures $\mu$ and $\tilde{\mu}$, without using the ordering relation between $\mu$ and $\tilde{\mu}$. Additionally, similar type of inequalities for the kernel type polynomials related to $\mu$ and $\tilde{\mu}$ are obtained.