  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 $-3\le\alpha\le0$, is a logarithmically convex
function of $r$ on $(0,1)$. We also show that the range $[-3,0]$ for $\alpha$
is best possible.
