Gaussian integral means of entire functions: logarithmic convexity and concavity

TL;DR

This paper characterizes when the $L^p$ integral means of entire functions, with respect to Gaussian measures, are logarithmically convex or concave, depending on parameters $p$ and $eta$.

Contribution

It provides a complete characterization of the convexity and concavity properties of Gaussian integral means for entire functions based on parameter conditions.

Findings

Determines conditions for logarithmic convexity of Gaussian integral means.

Determines conditions for logarithmic concavity of Gaussian integral means.

Establishes a clear parameter-dependent criterion for convexity and concavity.

Abstract

For $0<p<\infty$ and $\alpha\in (-\infty,\infty)$ we determine when the $L^p$ integral mean on $\{z\in\mathbb C: |z|\le r\}$ of an entire function with respect to the Gaussian area measure $e^{-\alpha|z|^2}\,dA(z)$ is logarithmic convex or logarithmic concave.