Logarithmic convexity of area integral means for analytic functions II

TL;DR

This paper proves that for certain weighted area measures, the $L^p$ integral means of analytic functions in the unit disk are logarithmically convex functions of the radius, extending previous results to new parameter ranges.

Contribution

It establishes the logarithmic convexity of area integral means for a broader class of weighted measures and parameters, advancing the understanding of their geometric properties.

Findings

Logarithmic convexity holds for $0<p< $ and $-2\, ext{to}\,0$ with respect to weighted area measures.

The result generalizes previous convexity results for unweighted or differently weighted measures.

Convexity is shown for the integral means as functions of the radius in the unit disk.

Abstract

For $0<p<\infty$ and $-2\le\alpha\le0$ we show that the $L^p$ integral mean on $rD$ of analytic function in the unit disk $D$ with respect to the weighted area measure $(1-|z|^2)^\alpha dA(z)$ is a logarithmically convex function of $r$ on $(0,1)$.