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Some Problems for Typically Real Functions

Published online by Cambridge University Press:  20 November 2018

James A. Jenkins
James A. Jenkins*
Affiliation:
Washington University, St. Louis
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Many extremal properties of the class of normalized univalent functions are shared by the class of typically real functions each considered in the unit circle. By the class T of typically real functions we mean those functions f(z), regular for |z| < 1 with f(0) = 0, f'(0) = 1, and such that f(z) > 0 for z > 0, (z) < 0 for > 0. This class was first studied by Rogosinski (4) who proved various simple properties for it. Later Robertson (3) took up the study and proved the following important representation result.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 299 - 304
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Gronwall, T. H., On the distortion in conformai mapping when the second coefficient in the mapping function has an assigned value, Proc. Nat. Acad. Sci. U.S.A., 6 (1920), 300–302.Google Scholar
2. Jenkins, James A., On a problem of Gronwall, Ann. Math., 59 (1954), 490–504.Google Scholar
3. Robertson, M. S., On the coefficients of a typically-real function, Bull. Amer. Math. Soc, 41 (1935), 565–572.Google Scholar
4. Rogosinski, W., Ueber positive harmonische Entwicklungen und typisch-reelle Potenzreihe, Math. Zeit, 35 (1932), 93–121.Google Scholar
5. Widder, D. V., The Laplace transform (Princeton University Press, 1946).Google Scholar