Logarithmic convexity of integral means for analytic functions

We show that the $L^2$ integral mean on $r\text{D}$ of an analytic function in the unit disk $\text{D}$ with respect to the weighted area measure $(1-\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{\alpha }dA(z)$, 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.