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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Image areas and $ H\sb{2}$ norms of analytic functions


Author: Shōji Kobayashi
Journal: Proc. Amer. Math. Soc. 91 (1984), 257-261
MSC: Primary 30D55; Secondary 30C80
MathSciNet review: 740181
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Abstract: For an analytic function $ f$ in the unit disc $ U$ with $ f(0) = 0$, the inequality $ \left\Vert f \right\Vert _2^2 \leqslant \frac{1}{\pi }$ area $ \{ f(U)\} $ is shown, where an equality occurs if and only if $ f$ is a constant multiple of an inner function. As a corollary, it is shown that for an analytic function in a general domain the square of its $ {H_2}$ norm is bounded by its Dirichlet integral, with the equality condition being settled


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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0740181-4
Article copyright: © Copyright 1984 American Mathematical Society