Logarithmic convexity of integral means for analytic functions

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 $(1-|z|^2)^α\,dA(z)$, where $-3\leα\le0$, is a logarithmically convex function of $r$ on $(0,1)$. We also show that the range $[-3,0]$ for $α$ is best possible.

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