Title: Log-Convexity of Weighted Area Integral Means of $H^p$ Functions on the Upper Half-plan,
Abstract: In the present work weighted area integral means $M_{p,\varphi}(f;{\mathrm
{Im}}z)$ are studied and it is proved that the function $y\to \log
M_{p,\varphi}(f;y)$ is convex in the case when $f$ belongs to a Hardy space on
the upper half-plane.