Logarithmic convexity of area integral means for analytic functions II

Abstract

For $0<p<\infty$ and $-2\le \alpha \le 0$ 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 $(0,1)$.

13 pages