Inequalities involving a measure of Marcellán class and zeros of corresponding orthogonal polynomials

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 Marcellán 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 Marcellán 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.

21 pages