Weighted Integral Means of Mixed Areas and Lengths under Holomorphic Mappings

TL;DR

This paper investigates the growth and convexity properties of weighted integral means of mixed areas and lengths under holomorphic mappings of the unit disk, revealing new monotonicity and convexity results.

Contribution

It introduces novel weighted integral means for mixed areas and lengths and establishes their monotonic growth and convexity properties under holomorphic maps.

Findings

Monotonic growth of weighted integral means

Logarithmic convexity of these means

Applicable to holomorphic functions from the unit disk

Abstract

This note addresses monotonic growths and logarithmic convexities of the weighted ($(1-t^2)^\alpha dt^2$, $-\infty<\alpha<\infty$, $0<t<1$) integral means $\mathsf{A}_{\alpha,\beta}(f,\cdot)$ and $\mathsf{L}_{\alpha,\beta}(f,\cdot)$ of the mixed area $(\pi r^2)^{-\beta}A(f,r)$ and the mixed length $(2\pi r)^{-\beta}L(f,r)$ ($0\le\beta\le 1$ and $0<r<1$) of $f(r\mathbb D)$ and $\partial f(r\mathbb D)$ under a holomorphic map $f$ from the unit disk $\mathbb D$ into the finite complex plane $\mathbb C$.