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

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DOI

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
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     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/}
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