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

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

Published online: 2014-03

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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 $\mathbb C$.

• Keywords

Monotonic growth logarithmic convexity mean mixed area mean mixed length isoperimetric inequality holomorphic map univalent function

32A10 32A36 51M25

@Article{ATA-30-1, author = {J. Xiao and W. Xu}, title = {Weighted Integral Means of Mixed Areas and Lengths Under Holomorphic Mappings}, journal = {Analysis in Theory and Applications}, year = {2014}, volume = {30}, number = {1}, pages = {1--19}, 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 $\mathbb C$.}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2014.v30.n1.1}, url = {http://global-sci.org/intro/article_detail/ata/4470.html} }
TY - JOUR T1 - Weighted Integral Means of Mixed Areas and Lengths Under Holomorphic Mappings AU - J. Xiao & W. Xu JO - Analysis in Theory and Applications VL - 1 SP - 1 EP - 19 PY - 2014 DA - 2014/03 SN - 30 DO - http://dor.org/10.4208/ata.2014.v30.n1.1 UR - https://global-sci.org/intro/article_detail/ata/4470.html KW - Monotonic growth KW - logarithmic convexity KW - mean mixed area KW - mean mixed length KW - isoperimetric inequality KW - holomorphic map KW - univalent function AB - 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 $\mathbb C$.