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The Existence of Quadratic Differentials in Simply Connected Regions of the Complex Plane

Published online by Cambridge University Press:  20 November 2018

E. Grassmann
E. Grassmann*
Affiliation:
University of Calgary, Calgary, Alberta
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Extract

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The general coefficient theorem [2] and the extended general coefficient theorem [3] state that the existence of certain quadratic differentials is a sufficient condition for a function to be a solution of certain extremum problems. The purpose of this paper is to show that in the case of simply connected regions this condition is also necessary.

We shall do this by a variational method of the Schiffer-Golusin-type. The main difficulty is, that the class of admissible functions for the general coefficient theorem is restricted and we must therefore have a method of variation with restrictions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 1 , February 1973 , pp. 83 - 91
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Golusin, G. M., GeometrischeFunktionentheorie(DeutscherVerlag der Wissenschaften, Berlin, 1957).Google Scholar
2. Jenkins, T. A., Univalent functions, 2nd Ed. (Springer, Berlin, 1965).Google Scholar
3. Jenkins, T. A., On certain extremal problems for the coefficients of univalent functions, J. Analyse Math. 18 (1967)Google Scholar
4. Lang, S., Analysis. I (Addison Wesley, Reading, 1968).Google Scholar