Approximations of generalized Bernstein functions

TL;DR

This paper derives sharp inequalities involving Beta and Gamma functions, aiding the approximation of generalized Bernstein functions with higher order Thorin-Bernstein functions, and explores properties of a related gamma function ratio.

Contribution

It introduces new sharp inequalities and properties related to Beta, Gamma, and generalized Bernstein functions, advancing approximation techniques.

Findings

Established sharp inequalities for Beta and Gamma functions.

Derived new properties of the gamma function ratio $x^{}\Gamma(x)/\Gamma(x+)$.

Enhanced approximation methods for generalized Bernstein functions.

Abstract

We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a related function, namely $x^{\lambda}\Gamma(x)/\Gamma(x+\lambda)$ are derived.