Functional inequalities for the Bickley function

TL;DR

This paper establishes new functional inequalities and monotonicity properties for the Bickley function using classical integral inequalities and monotone l'Hospital's rule, including the complete monotonicity of a related determinant function.

Contribution

It introduces novel complete monotonicity properties and inequalities for the Bickley function, expanding the understanding of its mathematical behavior.

Findings

Proved complete monotonicity of the Bickley function.

Derived new functional inequalities involving the Bickley function.

Established the complete monotonicity of a determinant function with Bickley entries.

Abstract

In this paper our aim is to deduce some complete monotonicity properties and functional inequalities for the Bickley function. The key tools in our proofs are the classical integral inequalities, like Chebyshev, H\"older-Rogers, Cauchy-Schwarz, Carlson and Gr\"uss inequalities, as well as the monotone form of l'Hospital's rule. Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Bickley function.