Log-Convexity of Weighted Area Integral Means of $H^p$ Functions on the Upper Half-plan

In the present work weighted area integral means $M_{p,\varphi}(f;{\mathrm {Im}}z)$arestudiedanditisprovedthatthefunction$y\to \log M_{p,\varphi}(f;y)$isconvexinthecasewhen$f$ belongs to a Hardy space on the upper half-plane.