Gaussian Integral Means of Entire Functions

TL;DR

This paper investigates Gaussian integral means of entire functions, deriving maximum principles, trace inequalities, and convexity properties related to these means, extending understanding of their behavior in complex analysis.

Contribution

It introduces a maximum principle for Gaussian integral means, establishes Fock-Sobolev trace inequalities, and proves convexity properties of these means in the logarithmic scale.

Findings

Maximum principle for ${ m M}_{p, ext{alpha}}(f,r)$ established

Fock-Sobolev trace inequalities derived

Convexity of ${ m M}_{p, ext{alpha}}(z^m,r)$ in $ ext{ln} r$ proven

Abstract

For an entire mapping $f:\mathbb C\mapsto\mathbb C$ and a triple $(p,\alpha, r)\in (0,\infty)\times(-\infty,\infty)\times(0,\infty]$, the Gaussian integral means of $f$ (with respect to the area measure $dA$) is defined by $$ {\mathsf M}_{p,\alpha}(f,r)=\Big({\int_{|z|<r}e^{-\alpha|z|^2}dA(z)}\Big)^{-1}{\int_{|z|<r}|f(z)|^p{e^{-\alpha|z|^2}}dA(z)}. $$ Via deriving a maximum principle for ${\mathsf M}_{p,\alpha}(f,r)$, we establish not only Fock-Sobolev trace inequalities associated with ${\mathsf M}_{p,p/2}(z^m f(z),\infty)$ (as $m=0,1,2,...$), but also convexities of $r\mapsto\ln {\mathsf M}_{p,\alpha}(z^m,r)$ and $r\mapsto {\mathsf M}_{2,\alpha<0}(f,r)$ in $\ln r$ with $0<r<\infty$.