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Area Integral Means of Analytic Functions in the Unit Disk

Published online by Cambridge University Press:  20 November 2018

Xiaohui Cui ,
Chunjie Wang  and
Kehe Zhu
Xiaohui Cui
Affiliation:
Department of Mathematics, Hebei University of Technology, Tianjin 300401, China e-mail: cxh_1124@126.com, wcj@hebut.edu.cn
Chunjie Wang
Affiliation:
Department of Mathematics, Hebei University of Technology, Tianjin 300401, China e-mail: cxh_1124@126.com, wcj@hebut.edu.cn
Kehe Zhu
Affiliation:
Kehe Zhu, Department of Mathematics, Shantou University, Guangdong 515063, China, e-mail : kzhu@math.albany.edu
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Abstract

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For an analytic function $f$ on the unit disk $\mathbb{D}$ , we show that the ${{L}^{2}}$ integral mean of $f$ on $\text{c}\,\text{}\,\text{ }\!\!|\!\!\text{ z }\!\!|\!\!\text{ }\,\text{}\,\text{r}$ with respect to the weighted area measure ${{\left( 1\,-\,|z{{|}^{2}} \right)}^{\alpha }}dA\left( z \right)$ is a logarithmically convex function of $r$ on $\left( c,\,1 \right)$ , where $-3\,\le \,\alpha \,\le \,0\,\text{and}\,\text{c}\,\in \,[\,0,\,1)$ . Moreover, the range $[-3,\,0]$ for $\alpha $ is best possible. When $c\,=\,0$ , our arguments here also simplify the proof for several results we obtained in earlier papers.

Keywords

30H1030H20logarithmic convexityarea integral meanBergman spaceHardy space

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 3 , 01 September 2018 , pp. 509 - 517
Copyright
Copyright © Canadian Mathematical Society 2018

References

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