Logarithmic convexity 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 $-3\le \alpha \le 0$, is a logarithmically convex function of $r$ on $(0,1)$. We also show that the range $[-3,0]$ for $\alpha$ is best possible.

Cited by in corpus (2)

• Weighted Integral Means of Mixed Areas and Lengths under Holomorphic Mappings
• Log-Convexity of Weighted Area Integral Means of $H^p$ Functions on the Upper Half-plan