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Schlicht Dirichlet Series

Published online by Cambridge University Press:  20 November 2018

M. S. Robertson
M. S. Robertson*
Affiliation:
Rutgers, the State University of New Jersey
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For power series

(1.1) for which

(1.2),

it has been known for four decades (1) that ƒ(z) is regular and univalent or schlicht in |z| < 1. This theorem, due to J. W. Alexander, has more recently been studied by Remak (5) who has shown that w = ƒ(z), under the hypothesis (1.2), maps |z| < 1 onto a star-like region, and if (1.2) is not satisfied=(z) need not be univalent in |z| < 1 for a proper choice of the amplitudes of the coefficients an.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958 , pp. 161 - 176
Copyright
Copyright © Canadian Mathematical Society 1958

References

1. Alexander, J. W., Functions which map the interior of the unit circle upon simple regions, Ann. of Math., 17 (1915), 12–22.Google Scholar
2. Goluzin, G. M., On distortion theorems and coefficients of univalent functions, Rec. Math. (Mat. Sbornik), N.S. (61), 19 (1946), 183-202.Google Scholar
3. Montel, P., Sur les fonctions localement univalentes ou multivalentes, Ann. Sci. École Norm. Sup. (3), 54 (1937), 39-54.Google Scholar
4. Noshiro, J., On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ. (1), 2 (1934), 129–155.Google Scholar
5. Remak, R., Ueber eine specielle Klasse schlichter konformer Abbildungen des Einheitskreises, Mathematica, Zutphen, 11 (1943), 175-192; 12 (1943), 43-49.Google Scholar