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LOGARITHMIC CONVEXITY OF AREA INTEGRAL MEANS FOR ANALYTIC FUNCTIONS II

Part of: Spaces and algebras of analytic functions

Published online by Cambridge University Press:  14 October 2014

CHUNJIE WANG ,
JIE XIAO  and
KEHE ZHU
CHUNJIE WANG
Affiliation:
Department of Mathematics, Hebei University of Technology, Tianjin 300401, China email wcj@hebut.edu.cn
JIE XIAO
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7, Canada email jxiao@mun.ca
KEHE ZHU*
Affiliation:
Department of Mathematics and Statistics, State University of New York, Albany, NY 12222, USA email kzhu@math.albany.edu
*
✉kzhu@math.albany.edu
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Abstract

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For $0<p<\infty$ and $-2\leq {\it\alpha}\leq 0$ we show that the $L^{p}$ integral mean on $r\mathbb{D}$ of an analytic function in the unit disk $\mathbb{D}$ with respect to the weighted area measure $(1-|z|^{2})^{{\it\alpha}}\,dA(z)$ is a logarithmically convex function of $r$ on $(0,1)$.

Keywords

logarithmic convexityarea integral meansHardy spaceBergman space

MSC classification

Primary: 30H10: Hardy spaces
Secondary: 30H20: Bergman spaces, Fock spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 1 , February 2015 , pp. 117 - 128
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

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Wang, C. and Zhu, K., ‘Logarithmic convexity of area integral means for analytic functions’, Math. Scand. 114 (2014), 149–160.CrossRefGoogle Scholar
Xiao, J. and Xu, W., ‘Weighted integral means of mixed areas and lengths under holomorphic mappings’, Anal. Theory Appl. 30 (2014), 1–19.Google Scholar
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