Upper bounds for Erd\'{e}lyi's multivariate Laguerre polynomials

TL;DR

This paper derives two new inequalities providing upper bounds for multivariate Laguerre polynomials, extending classical results and offering insights into their comparative bounds.

Contribution

It introduces two inequalities that generalize Szeg"o's inequality for multivariate Laguerre polynomials, expanding the theoretical understanding of their bounds.

Findings

Established two inequalities for multivariate Laguerre polynomials.

Generalized Szeg"o's inequality to multivariate case.

Provided comparative analysis of upper bounds.

Abstract

We establish in this paper two inequalities for the multivariate Laguerre polynomials introduced and studied by Arthur Erd\'{e}lyi [Sitzungsber. Akad. Wiss. Wien, Math.-Naturw. Kl., Abt. IIa 146 (1937), 431--467]. These inequalities generalize the well-known Szeg\"{o}'s inequality for the Laguerre polynomials $L_n^{(\alpha)}(x)$. We also mention briefly few insightful remarks giving a comparative analysis concerning the upper bounds of the derived inequalities in the concluding section.