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Derivatives and Length-Preserving Maps

Published online by Cambridge University Press:  20 November 2018

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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Let a be a constant, |a| = 1. We shall prove meromorphic (M) and bounded-holomorphic (BH) versions of the following prototype: (P) Let f and g be holomorphic in a domain D. Then, |f'| = |g'| in D if and only if there exist constant a, b with f = ag + b in D. (M) Let f and g be meromorphic in D. Then, |f'|/(1 + |f|2) = |g'|/(1 + |g|2) in D if and only if there exist a, b with |b| ≦ ∞ such that f = a(g - b)/(\ + g). (BH) Let f and g be holomorphic and bounded, |f| < 1, |g| < 1, in D. Then, |f'|/ (1 - |f|2) = |g'|/(1 - |g|2) in D if and only if there exist a, b with |b| < 1, such that f = a(g - b)/(1 - g).

Keywords

Primary, 30A99Secondary30C45
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 3 , 01 September 1987 , pp. 379 - 384
Copyright
Copyright © Canadian Mathematical Society 1987

References

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