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

TL;DR

This paper investigates the convexity properties of weighted area integral means of Hardy space functions on the upper half-plane, establishing a log-convexity result for these means.

Contribution

It proves the log-convexity of weighted area integral means for Hardy space functions on the upper half-plane, extending understanding of their geometric properties.

Findings

Proved log-convexity of weighted area integral means for Hardy functions.

Established convexity of the logarithm of these means with respect to the imaginary part.

Extended classical results to weighted integral means in the upper half-plane.

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.