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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


An integral criterion for normal functions

Authors: Rauno Aulaskari and Peter Lappan
Journal: Proc. Amer. Math. Soc. 103 (1988), 438-440
MSC: Primary 30D45
MathSciNet review: 943062
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Abstract: A new characterization for normal functions is given. It is shown that a function $ f$ meromorphic in the unit disk is a normal function if and only if for each $ \delta > 0$ and each $ p > 2$ there exists a constant $ {K_f}(\delta ,p)$ such that, for each hyperbolic disk $ \Omega $ with hyperbolic radius $ \delta $,

$\displaystyle \iint_\Omega {{{(1 - {{\left\vert z \right\vert}^2})}^{p - 2}}{{({f^ \ne }(z))}^p}dA(z) \leq {K_f}(\delta ,p)},$

where $ {f^ \ne }(z)$ denotes the spherical derivative of $ f$ and $ dA(z)$ is the Euclidean element of area. It is shown by example that this characterization is not valid for $ p = 2$.

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PII: S 0002-9939(1988)0943062-X
Article copyright: © Copyright 1988 American Mathematical Society