Gaussian integral means of entire functions: logarithmic convexity and concavity

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.Comment: 9 page

Gaussian Integral Means of Entire Functions

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 $0<r<\infty$.Comment: 18 page