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

Li, Haiying, and Liu, Taotao. "Convexities of Gaussian integral means and weighted integral means for analytic functions." Czechoslovak Mathematical Journal 69.2 (2019): 525-543. <http://eudml.org/doc/294430>.

@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 -