TY - JOUR
T1 - Weighted Integral Means of Mixed Areas and Lengths Under Holomorphic Mappings
JO - Analysis in Theory and Applications
VL - 1
SP - 1
EP - 19
PY - 2014
DA - 2014/03
SN - 30
DO - http://doi.org/10.4208/ata.2014.v30.n1.1
UR - https://global-sci.org/intro/article_detail/ata/4470.html
KW - Monotonic growth, logarithmic convexity, mean mixed area, mean mixed length, isoperimetric inequality, holomorphic map, univalent function.
AB - <p style="text-align: justify;">This note addresses monotonic growths and logarithmic convexities of the weighted ($(1-t^2)^\alpha dt^2$, $-\infty &lt;\alpha &lt;\infty$, $0&lt; t&lt; 1$) 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 $0&lt; r&lt; 1$) 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 $\mathbb C$.</p>