Inequalities for the one-dimensional analogous of the Coulomb potential

TL;DR

This paper investigates the mathematical properties of a one-dimensional regularization of the Coulomb potential, deriving inequalities and bounds that have implications for atomic physics and probability theory.

Contribution

It introduces new monotonicity, convexity, and Turán-type inequalities for the regularized Coulomb potential, leading to improved bounds on the Mills ratio of the normal distribution.

Findings

Established monotonicity and convexity properties of the potential.

Derived Turán-type inequalities for the function.

Provided new tight upper bounds for the Mills ratio.

Abstract

In this paper our aim is to present some monotonicity and convexity properties for the one dimensional regularization of the Coulomb potential, which has applications in the study of atoms in magnetic fields and which is in fact a particular case of the Tricomi confluent hypergeometric function. Moreover, we present some Tur\'an type inequalities for the function in the question and we deduce from these inequalities some new tight upper bounds for the Mills ratio of the standard normal distribution.