Geodesic
Area Integral Means of Analytic Functions in the Unit Disk
Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 509-517

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2017-053-3
Mots-clés : 30H10, 30H20, logarithmic convexity, area integral mean, Bergman space, Hardy space 
Cui, Xiaohui; Wang, Chunjie; Zhu, Kehe. Area Integral Means of Analytic Functions in the Unit Disk. Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 509-517. doi: 10.4153/CMB-2017-053-3
@article{10_4153_CMB_2017_053_3,
     author = {Cui, Xiaohui and Wang, Chunjie and Zhu, Kehe},
     title = {Area {Integral} {Means} of {Analytic} {Functions} in the {Unit} {Disk}},
     journal = {Canadian mathematical bulletin},
     pages = {509--517},
     year = {2018},
     volume = {61},
     number = {3},
     doi = {10.4153/CMB-2017-053-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-053-3/}
}
TY  - JOUR
AU  - Cui, Xiaohui
AU  - Wang, Chunjie
AU  - Zhu, Kehe
TI  - Area Integral Means of Analytic Functions in the Unit Disk
JO  - Canadian mathematical bulletin
PY  - 2018
SP  - 509
EP  - 517
VL  - 61
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-053-3/
DO  - 10.4153/CMB-2017-053-3
ID  - 10_4153_CMB_2017_053_3
ER  - 
%0 Journal Article
%A Cui, Xiaohui
%A Wang, Chunjie
%A Zhu, Kehe
%T Area Integral Means of Analytic Functions in the Unit Disk
%J Canadian mathematical bulletin
%D 2018
%P 509-517
%V 61
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-053-3/
%R 10.4153/CMB-2017-053-3
%F 10_4153_CMB_2017_053_3

[1] [1] Duren, P., Theory of H3 Spaces. Pure and Applied Mathematics, 30, Academic Press, New York-London, 1970. Google Scholar

[2] [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/100 Google Scholar

[3] [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/10.1007/978-1-4612-0497-8 Google Scholar

[4] [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-x Google Scholar

[5] [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-x Google Scholar

[6] [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/S1446788714000457 Google Scholar

[7] [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-16643 Google Scholar

[8] [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] [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/S0002-9939-2010-10797-9 Google Scholar

Cité par Sources :