Convexities of Gaussian integral means and weighted integral means for analytic functions

@article{Li2019,
abstract = {We first show that the Gaussian integral means of $f\colon \mathbb \{C\}\rightarrow \mathbb \{C\}$ (with respect to the area measure $\{\rm e\}^\{-\alpha |z|^\{2\}\} \{\rm d\} A(z)$) is a convex function of $r$ on $(0,\infty )$ when $\alpha \le 0$. We then prove that the weighted integral means $A_\{\alpha ,\beta \}(f,r)$ and $L_\{\alpha ,\beta \}(f,r)$ of the mixed area and the mixed length of $f(r\mathbb \{D\})$ and $\partial f(r\mathbb \{D\})$, respectively, also have the property of convexity in the case of $\alpha \le 0$. Finally, we show with examples that the range $\alpha \le 0$ is the best possible.},
author = {Li, Haiying, Liu, Taotao},
journal = {Czechoslovak Mathematical Journal},
keywords = {Gaussian integral means; weighted integral means; analytic function; convexity},
language = {eng},
number = {2},
pages = {525-543},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Convexities of Gaussian integral means and weighted integral means for analytic functions},
url = {http://eudml.org/doc/294430},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Li, Haiying
AU - Liu, Taotao
TI - Convexities of Gaussian integral means and weighted integral means for analytic functions
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 2
SP - 525
EP - 543
AB - We first show that the Gaussian integral means of $f\colon \mathbb {C}\rightarrow \mathbb {C}$ (with respect to the area measure ${\rm e}^{-\alpha |z|^{2}} {\rm d} A(z)$) is a convex function of $r$ on $(0,\infty )$ when $\alpha \le 0$. We then prove that the weighted integral means $A_{\alpha ,\beta }(f,r)$ and $L_{\alpha ,\beta }(f,r)$ of the mixed area and the mixed length of $f(r\mathbb {D})$ and $\partial f(r\mathbb {D})$, respectively, also have the property of convexity in the case of $\alpha \le 0$. Finally, we show with examples that the range $\alpha \le 0$ is the best possible.
LA - eng
KW - Gaussian integral means; weighted integral means; analytic function; convexity
UR - http://eudml.org/doc/294430
ER -