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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Volume integral means of holomorphic functions


Authors: Jie Xiao and Kehe Zhu
Journal: Proc. Amer. Math. Soc. 139 (2011), 1455-1465
MSC (2010): Primary 32A10, 32A36, 32A35, 51M25
Published electronically: November 18, 2010
MathSciNet review: 2748439
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Abstract: The classical integral means of a holomorphic function $ f$ in the unit disk are defined by

$\displaystyle \Bigg[\frac1{2\pi}\int_0^{2\pi}\vert f(re^{i\theta})\vert^p d\theta\Bigg]^{1/p}, \quad 0\le r<1.$

These integral means play an important role in modern complex analysis. In this note we consider integral means of holomorphic functions in the unit ball $ \mathbb{B}_n$ in $ \mathbb{C}^n$ with respect to weighted volume measures,

$\displaystyle M_{p,\alpha}(f,r)=\left[\frac{1}{v_\alpha(r\mathbb{B}_n)} \int_{r\mathbb{B}_n}\vert f(z)\vert^p dv_\alpha(z)\right]^{1/p}, \quad 0\le r<1,$

where $ \alpha$ is real, $ dv_\alpha(z)=(1-\vert z\vert^2)^\alpha dv(z)$, and $ dv$ is volume measure on $ \mathbb{B}_n$. We show that $ M_{p,\alpha}(f,r)$ increases with $ r$ strictly unless $ f$ is a constant, but in contrast with the classical case, $ \log M_{p,\alpha}(f,r)$ is not always convex in $ \log r$. As an application, we show that if $ \alpha\le-1$, $ M_{p,\alpha}(f,r)$ is bounded in $ r$ if and only if $ f$ belongs to the Hardy space $ H^p$, while if $ \alpha>-1$, $ M_{p,\alpha}(f,r)$ is bounded in $ r$ if and only if $ f$ is in the weighted Bergman space $ A^p_\alpha$.


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Additional Information

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, New York 12222
Email: kzhu@albany.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10797-9
PII: S 0002-9939(2010)10797-9
Keywords: Volume/area integral means, monotonicity, logarithmic convexity, Bergman/Hardy spaces, isoperimetric-type inequalities, weighted Ricci curvatures.
Received by editor(s): May 3, 2010
Published electronically: November 18, 2010
Additional Notes: The first author was supported in part by NSERC of Canada
Communicated by: Richard Rochberg
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.