Logarithmic Convexity of Area Integral Means for Analytic Functions - Wang, Chunjie; Zhu, Kehe (MATHEMATICA SCANDINAVICA, 2014)
We show that the $L^2$ integral mean on $r\mathsf{D}$ of an analytic function in the unit disk $\mathsf{D}$ 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.