Weighted Integral Means of Mixed Areas and Lengths under Holomorphic Mappings

Abstract

This note addresses monotonic growths and logarithmic convexities of the weighted ($(1-t^2{)}^{\alpha }d{t}^2$, $-\infty <\alpha <\infty$, $0<t<1$) 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 $\mathbb{C}$.

References in corpus (1)

• Logarithmic convexity of integral means for analytic functions