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Area Integral Means of Analytic Functions in the Unit Disk

• Xiaohui Cui,
Department of Mathematics, Hebei University of Technology, Tianjin 300401, China
• Chunjie Wang,
Department of Mathematics, Hebei University of Technology, Tianjin 300401, China
• Kehe Zhu,
Department of Mathematics, Shantou University, Guangdong 515063, China
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Abstract

For an analytic function $f$ on the unit disk $\mathbb D$ we show that the $L^2$ integral mean of $f$ on $c\lt |z|\lt r$ with respect to the weighted area measure $(1-|z|^2)^\alpha\,dA(z)$ is a logarithmically convex function of $r$ on $(c,1)$, where $-3\le\alpha\le0$ and $c\in[0,1)$. 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: logarithmic convexity, area integral mean, Bergman space, Hardy space
 MSC Classifications: 30H10 - Hardy spaces 30H20 - Bergman spaces, Fock spaces

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