Volume Integral Mean of Holomorphic Function on Polydisc

Lijuan Xu; Hua Liu; Juan Chen; Xiaoli Bian

doi:doi:10.11648/j.ijtam.20200601.12

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Volume Integral Mean of Holomorphic Function on Polydisc

Lijuan Xu, Hua Liu, Juan Chen, Xiaoli Bian

Published in International Journal of Theoretical and Applied Mathematics (Volume 6, Issue 1)

Received: 17 December 2019 Accepted: 4 January 2020 Published: 13 January 2020

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Abstract

Let f be an analytic function in the Hardy space on the polydisc P₂. In this article we discuss the area integral means M_p (f, r) of f on the polydisc P₂ with radius r, and its weighted volume means M_p,α (f, r) with to the weight (1-|z₁|²)^a×(1-|z₂|²)^a. We prove that both M_p (f, r) and M_p,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 M_p,α (f, r) is not always convex with respect to log r, although that we still prove that log M_p (f, r) is logarithmically convex.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 6, Issue 1)
DOI	10.11648/j.ijtam.20200601.12
Page(s)	14-18
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Hardy Space, Polydisc, Integral Means, Logarithmically Convex

References

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[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).
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[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.
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[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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APA Style

Lijuan Xu, Hua Liu, Juan Chen, Xiaoli Bian. (2020). Volume Integral Mean of Holomorphic Function on Polydisc. International Journal of Theoretical and Applied Mathematics, 6(1), 14-18. https://doi.org/10.11648/j.ijtam.20200601.12

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ACS Style

Lijuan Xu; Hua Liu; Juan Chen; Xiaoli Bian. Volume Integral Mean of Holomorphic Function on Polydisc. Int. J. Theor. Appl. Math. 2020, 6(1), 14-18. doi: 10.11648/j.ijtam.20200601.12

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AMA Style

Lijuan Xu, Hua Liu, Juan Chen, Xiaoli Bian. Volume Integral Mean of Holomorphic Function on Polydisc. Int J Theor Appl Math. 2020;6(1):14-18. doi: 10.11648/j.ijtam.20200601.12

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@article{10.11648/j.ijtam.20200601.12,
  author = {Lijuan Xu and Hua Liu and Juan Chen and Xiaoli Bian},
  title = {Volume Integral Mean of Holomorphic Function on Polydisc},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {6},
  number = {1},
  pages = {14-18},
  doi = {10.11648/j.ijtam.20200601.12},
  url = {https://doi.org/10.11648/j.ijtam.20200601.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20200601.12},
  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.},
 year = {2020}
}

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TY  - JOUR
T1  - Volume Integral Mean of Holomorphic Function on Polydisc
AU  - Lijuan Xu
AU  - Hua Liu
AU  - Juan Chen
AU  - Xiaoli Bian
Y1  - 2020/01/13
PY  - 2020
N1  - https://doi.org/10.11648/j.ijtam.20200601.12
DO  - 10.11648/j.ijtam.20200601.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 14
EP  - 18
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20200601.12
AB  - 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.
VL  - 6
IS  - 1
ER  -

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Author Information

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

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APA Style

Lijuan Xu, Hua Liu, Juan Chen, Xiaoli Bian. (2020). Volume Integral Mean of Holomorphic Function on Polydisc. International Journal of Theoretical and Applied Mathematics, 6(1), 14-18. https://doi.org/10.11648/j.ijtam.20200601.12

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ACS Style

Lijuan Xu; Hua Liu; Juan Chen; Xiaoli Bian. Volume Integral Mean of Holomorphic Function on Polydisc. Int. J. Theor. Appl. Math. 2020, 6(1), 14-18. doi: 10.11648/j.ijtam.20200601.12

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AMA Style

Lijuan Xu, Hua Liu, Juan Chen, Xiaoli Bian. Volume Integral Mean of Holomorphic Function on Polydisc. Int J Theor Appl Math. 2020;6(1):14-18. doi: 10.11648/j.ijtam.20200601.12

Copy | Download

@article{10.11648/j.ijtam.20200601.12,
  author = {Lijuan Xu and Hua Liu and Juan Chen and Xiaoli Bian},
  title = {Volume Integral Mean of Holomorphic Function on Polydisc},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {6},
  number = {1},
  pages = {14-18},
  doi = {10.11648/j.ijtam.20200601.12},
  url = {https://doi.org/10.11648/j.ijtam.20200601.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20200601.12},
  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.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Volume Integral Mean of Holomorphic Function on Polydisc
AU  - Lijuan Xu
AU  - Hua Liu
AU  - Juan Chen
AU  - Xiaoli Bian
Y1  - 2020/01/13
PY  - 2020
N1  - https://doi.org/10.11648/j.ijtam.20200601.12
DO  - 10.11648/j.ijtam.20200601.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 14
EP  - 18
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20200601.12
AB  - 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.
VL  - 6
IS  - 1
ER  -

Copy | Download