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Volume Integral Mean of Holomorphic Function on Polydisc

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 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.

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
[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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  • 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
  • College of Science, Tianjin University of Technology and Education, Tianjin, China

  • College of Science, Tianjin University of Technology and Education, Tianjin, China

  • Basic Courses Department, Tianjin Sino-German University of applied Science, Tianjin, China

  • College of Science, Tianjin University of Technology and Education, Tianjin, China

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