Volume 30, Issue 1
Weighted Integral Means of Mixed Areas and Lengths Under Holomorphic Mappings

J. Xiao and W. Xu


Anal. Theory Appl., 30 (2014), pp. 1-19

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  • Abstract

This note addresses monotonic growths and logarithmic convexities ofthe 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 $(\pir^2)^{-\beta}A(f,r)$ 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 $\partialf(r\mathbb D)$ under a holomorphic map $f$ from the unit disk$\mathbb D$ into the finite complex plane $\mathbb C$.

  • History

Published online: 2014-03

  • AMS Subject Headings

32A10, 32A36, 51M25

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