Inequalities involving a measure of Marcell\'{a}n 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 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.