Tur\'an type inequalities for Kr\"atzel functions

TL;DR

This paper investigates Turán type inequalities and complete monotonicity properties of Kr"atzel functions, providing new inequalities and monotonicity results for this class of special functions.

Contribution

The paper establishes Turán type inequalities and complete monotonicity results for Kr"atzel functions, extending the understanding of their mathematical properties.

Findings

Proved Turán type inequalities for Kr"atzel functions.

Established complete monotonicity of certain determinant functions involving Kr"atzel functions.

Extended the theory of inequalities for special functions.

Abstract

Complete monotonicity, Laguerre and Tur\'an type inequalities are established for the so-called Kr\"atzel function $Z_{\rho}^{\nu},$ defined by $$Z_{\rho}^{\nu}(u)=\int_0^{\infty}t^{\nu-1}e^{-t^{\rho}-\frac{u}{t}}\dt,$$ where $u>0$ and $\rho,\nu\in\mathbb{R}.$ Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Kr\"atzel function.