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# Normal Functions: $L^p$ Estimates

Published:1997-02-01
Printed: Feb 1997
• Huaihui Chen
• Paul M. Gauthier
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## Abstract

For a meromorphic (or harmonic) function $f$, let us call the dilation of $f$ at $z$ the ratio of the (spherical) metric at $f(z)$ and the (hyperbolic) metric at $z$. Inequalities are known which estimate the $\sup$ norm of the dilation in terms of its $L^p$ norm, for $p>2$, while capitalizing on the symmetries of $f$. In the present paper we weaken the hypothesis by showing that such estimates persist even if the $L^p$ norms are taken only over the set of $z$ on which $f$ takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that $p=2$.
 MSC Classifications: 30D45 - Bloch functions, normal functions, normal families 30F35 - Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx]