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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

1. Gackstatter, F., Die Gausssche und mittlere Kriimmung der Realteilflächen in der Thorie der meromorphen Funktionen, Math. Nachr. 54 (1972), pp. 211—227.Google Scholar
2. Kreyszig, E., Die Realteil- und Imaginärteilflächen analytischer Funktionen, Elemente der Mathematik 24(1969), pp. 25–31.Google Scholar
3. Kreyszig, E. and Pendl, A., Über die Gauss-Krümmung der Real- und Imaginärteilflächen analytischer Funktionen, Elemente der Mathematik 28 (1973), pp. 10–13.Google Scholar
4. Jerrard, R., Curvatures of surfaces associated with holomorphic functions, Colloquium Math. 21 (1970), pp. 127–132.Google Scholar
5. Tsuji, M., Potential theory in modern function theory, Maruzen, Tokyo, 1959.Google Scholar