Gaussian Integral Means of Entire Functions

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 $$ {\mathsf M}<em>{p,\alpha}(f,r)=\Big({\int</em>{|z|&lt;r}e^{{-\alpha|z|^{2}dA(z)}\Big)^{{-1}{\int_{|z|&lt;r}|f(z)|^{p{e^{{-\alpha|z|^2}}dA(z)}.}}}}} $$ Via deriving a maximum principle for ${\mathsf M}<em>{p,\alpha}(f,r)$, we establish not only Fock-Sobolev trace inequalities associated with ${\mathsf M}</em>{p,p/2}(z^m f(z),\infty)$ (as $m=0,1,2,...$), but also convexities of $r\mapsto\ln {\mathsf M}<em>{p,\alpha}(z<sup>m,r)$ and $r\mapsto {\mathsf M}</em>{2,\alpha&lt;0}(f,r)$ in $\ln r$ with $0&lt;r&lt;\infty$.