Logarithmic convexity of integral means for analytic functions

TL;DR

This paper proves that the $L^2$ integral mean of analytic functions in the unit disk, weighted by $(1-|z|^2)^eta$, is logarithmically convex in the radius for $eta$ between -3 and 0, and this range is optimal.

Contribution

It establishes the logarithmic convexity of weighted $L^2$ integral means for a specific range of weights, extending known results and proving the optimality of this range.

Findings

Logarithmic convexity holds for weights with $eta$ in [-3,0].

Range [-3,0] for $eta$ is proven to be optimal.

The result extends the understanding of integral means for analytic functions.

Abstract

We show that the $L^2$ integral mean on $r\D$ of an analytic function in the unit disk $\D$ with respect to the weighted area measure $(1-|z|^2)^\alpha\,dA(z)$, where $-3\le\alpha\le0$, is a logarithmically convex function of $r$ on $(0,1)$. We also show that the range $[-3,0]$ for $\alpha$ is best possible.