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Length and area inequalities for the derivative of a bounded and holomorphic function

Published online by Cambridge University Press:  17 April 2009

Shinji Yamashita
Shinji Yamashita
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, Fukasawa, Setagaya, Tokyo 158, Japan.
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Abstract

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The Schwarz-Pick lemma,

for f analytic and bounded, |f|<1, in the disk |z|<1, is refined:

where Φ(z, r) is a quantity determined by the non-Euclidean area of the image of

and ψ(z, r) is that determined by the non-Euclidean length of the image of the boundary of D(z, r). The multiplicities in both images by f are not counted.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 30 , Issue 3 , December 1984 , pp. 457 - 462
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Collingwood, E.F. and Lohwater, A.J., The theory of cluster sets (Cambridge University Press, Cambridge, 1966).Google Scholar
[2]Dufresnoy, J., “Sur l'aire sphérique décrite par les valeurs d'une fonction méromorphe”, Bull. Sci. Math. 65 (1941), 214–219.Google Scholar
[3]Osserman, R., “The isoperimetric inequality”, Bull. Amer. Math. Soc. 84 (1978), 1182–1238.Google Scholar