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Area Integral Means of Analytic Functions in the Unit Disk

Published online by Cambridge University Press:  20 November 2018

Xiaohui Cui ,
Chunjie Wang  and
Kehe Zhu
Xiaohui Cui
Affiliation:
Department of Mathematics, Hebei University of Technology, Tianjin 300401, China e-mail: cxh_1124@126.com, wcj@hebut.edu.cn
Chunjie Wang
Affiliation:
Department of Mathematics, Hebei University of Technology, Tianjin 300401, China e-mail: cxh_1124@126.com, wcj@hebut.edu.cn
Kehe Zhu
Affiliation:
Kehe Zhu, Department of Mathematics, Shantou University, Guangdong 515063, China, e-mail : kzhu@math.albany.edu
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Abstract

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For an analytic function $f$ on the unit disk $\mathbb{D}$, we show that the ${{L}^{2}}$ integral mean of $f$ on $\text{c}\,\text{}\,\text{ }\!\!|\!\!\text{ z }\!\!|\!\!\text{ }\,\text{}\,\text{r}$ with respect to the weighted area measure ${{\left( 1\,-\,|z{{|}^{2}} \right)}^{\alpha }}dA\left( z \right)$ is a logarithmically convex function of $r$ on $\left( c,\,1 \right)$, where $-3\,\le \,\alpha \,\le \,0\,\text{and}\,\text{c}\,\in \,[\,0,\,1)$. Moreover, the range $[-3,\,0]$ for $\alpha $ is best possible. When $c\,=\,0$, our arguments here also simplify the proof for several results we obtained in earlier papers.

Keywords

30H1030H20logarithmic convexityarea integral meanBergman spaceHardy space
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 3 , 01 September 2018 , pp. 509 - 517
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Duren, P., Theory of H3 Spaces. Pure and Applied Mathematics, 30, Academic Press, New York-London, 1970.Google Scholar
[2] Duren, P. and Schuster, A., Bergman Spaces. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2005. http://dx.doi.Org/10.1090/surv/100Google Scholar
[3] Hedenmalm, H., Korenblum, B., and Zhu, K., Theory of Bergman Spaces. Graduate Texts in Mathematics, 199, Springer-Verlag, New York, 2000. http://dx.doi.Org/10.1007/978-1-4612-0497-8Google Scholar
[4] Wang, C. and Xiao, J., Gaussian integral means of entire functions. Complex Anal. Oper. Theory 8(2014), 1487–1505. http://dx.doi.Org/10.1007/s11785-013-0339-xGoogle Scholar
[5] Wang, C. and Xiao, J., Addendum to “Gaussian integral means of entire functions”. Complex Anal. Oper. Theory 10(2016), 495–503. http://dx.doi.Org/10.1007/s11785-015-0447-xGoogle Scholar
[6] Wang, C., Xiao, J., and Zhu, K., Logarithmic convexity ofarea integral means for analytic functions II. J. Aust. Math. Soc. 98(2015), 117–128. http://dx.doi.Org/10.1017/S1446788714000457Google Scholar
[7] Wang, C. and Zhu, K., Logarithmic convexity ofarea integral means for analytic functions. Math. Scand. 114(2014), 149–160. http://dx.doi.org/10.7146/math.scand.a-16643Google Scholar
[8] 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
[9] Xiao, J. and Zhu, K., Volume integral means of holomorphic functions. Proc. Amer. Math. Soc. 139(2011), 1455–1465. http://dx.doi.org/10.1090/S0002-9939-2010-10797-9Google Scholar