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Logarithmic Convexity of Area Integral Means for Analytic Functions

Chunjie Wang, Kehe Zhu

Abstract


We show that the $L^2$ integral mean on $r\mathsf{D}$ of an analytic function in the unit disk $\mathsf{D}$ with respect to the weighted area measure $(1-|z|^2)^\alpha\,dA(z)$, where $-3\le\alpha\le0$, is a logarithmically convex function of $r$ on $(0,1)$. We also show that the range $[-3,0]$ for $\alpha$ is best possible.

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DOI: http://dx.doi.org/10.7146/math.scand.a-16643

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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