Weighted Integral Means of Mixed Areas and Lengths under Holomorphic Mappings

Abstract

This note addresses monotonic growths and logarithmic convexities of the weighted ($(1-t^2)^αdt^2$, $-\infty<α<\infty$, $0<t<1$) integral means $\mathsf{A}_{α,β}(f,\cdot)$ and $\mathsf{L}_{α,β}(f,\cdot)$ of the mixed area $(πr^2)^{-β}A(f,r)$ and the mixed length $(2πr)^{-β}L(f,r)$ ($0\leβ\le 1$ 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 $\mathbb C$.