Volume 6, Issue 1, February 2020, Page: 14-18
Volume Integral Mean of Holomorphic Function on Polydisc
Lijuan Xu, College of Science, Tianjin University of Technology and Education, Tianjin, China
Hua Liu, College of Science, Tianjin University of Technology and Education, Tianjin, China
Juan Chen, Basic Courses Department, Tianjin Sino-German University of applied Science, Tianjin, China
Xiaoli Bian, College of Science, Tianjin University of Technology and Education, Tianjin, China
Received: Dec. 17, 2019;       Accepted: Jan. 4, 2020;       Published: Jan. 13, 2020
DOI: 10.11648/j.ijtam.20200601.12      View  522      Downloads  185
Abstract
Let f be an analytic function in the Hardy space on the polydisc P2. In this article we discuss the area integral means Mp (f, r) of f on the polydisc P2 with radius r, and its weighted volume means Mp,α (f, r) with to the weight (1-|z1|2)a×(1-|z2|2)a. We prove that both Mp (f, r) and Mp,α (f, r) are strictly increasing in r unless f is a constant. In contrast to the classical case, we also give a example to show that log Mp,α (f, r) is not always convex with respect to log r, although that we still prove that log Mp (f, r) is logarithmically convex.
Keywords
Hardy Space, Polydisc, Integral Means, Logarithmically Convex
To cite this article
Lijuan Xu, Hua Liu, Juan Chen, Xiaoli Bian, Volume Integral Mean of Holomorphic Function on Polydisc, International Journal of Theoretical and Applied Mathematics. Vol. 6, No. 1, 2020, pp. 14-18. doi: 10.11648/j.ijtam.20200601.12
Copyright
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
Zhu K. H. Space of Holomorphic Functions in the Unit ball [M]. New York: Springer-verlag, 2005.
[2]
Xiao J, Zhu K. H. Volume integral means of holomorphic functions [J]. Proc. American Mathematical Society, 2011, 139 (4): 1455-1465.
[3]
Liu Hua. On the linear extreme of bergman space on polydisc [J]. Of Xuzhou Normal Uni: Natural Sciences, 2003, 21 (2): 1-4. (Chinese).
[4]
Duren P. Theory of Hp Space [M]. New York: Academic Press, 1970.
[5]
Shi Jihuai. Foundations of function theory of several complex variables [M]. Bei Jing: Higher Education Press, 1996. (Chinese).
[6]
Taylor A E. New proofs of some theorems of Hardy by Banach space methods [J], Math. Magazine, 23 (1950), 115-124.
[7]
Zhu K. H. (2004). Translating Inequalities between Hardy and Bergman Spaces [J]. American Mathematical Monthly. 111. 10.2307/4145071.
[8]
Zhu K. H. (1990). On certain unitary operators and composition operators [J]. 10.1090/pspum/051.2/1077459.
[9]
Zhu K. H. (2005). Spaces of Holomorphic Functions in the Unit Ball [J]. Grad Texts in Math. 10.1007/0-387-27539-8.
[10]
Zhu K. H. (2015). Singular Integral Operators on the Fock Space. Integral Equations and Operator Theory. 81. 10.1007/s00020-015-2222-9.
[11]
Xiao Jie. (2019). Prescribing Capacitary Curvature Measures on Planar Convex Domains [J]. The Journal of Geometric Analysis. 10.1007/s12220-019-00180-9.
[12]
Xiao Jie. (2015). On the variational $p$-capacity problem in the plane [J]. Communications on Pure and Applied Analysis. 14. 959-968. 10.3934/cpaa.2015.14.959.
[13]
Xiao Jie. (2017). The p-Affine Capacity Redux [J]. The Journal of Geometric Analysis. 27. 10.1007/s12220-017-9785-4.
[14]
Xiao Jie. (2015). The $$p$$ p -Affine Capacity [J]. The Journal of Geometric Analysis. 26. 10.1007/s12220-015-9579-5.
[15]
Xiao, Jie. (2014). Optimal geometric estimates for fractional Sobolev capacities [J]. Comptes Rendus Mathematique. 354. 10.1016/j.crma.2015.10.014.
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