Czechoslovak Mathematical Journal, Vol. 69, No. 2, pp. 525-543, 2019


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

Haiying Li, Taotao Liu

Received September 18, 2017.   Published online August 24, 2018.

Abstract:  We first show that the Gaussian integral means of $f \mathbb{C}\to\mathbb{C}$ (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 $\alpha\leq0$. 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 $\alpha\leq0$. Finally, we show with examples that the range $\alpha\leq0$ is the best possible.
Keywords:  Gaussian integral means; weighted integral means; analytic function; convexity
Classification MSC:  30H10, 30H20
DOI:  10.21136/CMJ.2018.0432-17


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Affiliations:   Haiying Li, Taotao Liu, School of Mathematics and Information Science, Henan Normal University, 46# East of Construction Road, Xinxiang, Henan, 453007 P. R. China, e-mail: haiyingli2012@yahoo.com


 
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