ON A CONVEXITY THEOREM OF RUSKAI AND WERNER AND RELATED RESULTS

Abstract

We show that the function \[ V_q(x)=\frac{2e^{x^2}}{\Gamma(q+1)}\int_{x}^{\infty}e^{-t^2}(t^2-x^2)^qdt\quad{(-1<q\in\mathbf{R}; 0<x\in \mathbf{R})}, \] which has applications in the study of atoms in magnetic fields, satisfies certain monotonicity and convexity properties as well as inequalities. In particular, we prove that $1/V_q$ is convex on $(0,\infty)$ if and only if $q\geq 0$. This extends a recent result of M. B. Ruskai and E. Werner, who established the convexity for all integers $q\geq 0$.