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The Valence of Sums and Products

Published online by Cambridge University Press:  20 November 2018

A. W. Goodman
A. W. Goodman*
Affiliation:
University of South Florida, Tampa, Florida
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Extract

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A function ƒ(z) is said to be p-valent in a region if it is regular in if the equation

1

has p distinct roots in for some particular w0 , and if for each complex w0 , equation (1) does not have more than p roots in . The function ƒ(z) is also said to have valence p in . In the case when p = 1, the function is said to be univalent in .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1173 - 1177
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

This work was supported by the National Science Foundation, Research Grant GP-5689.

References

1. Goodman, A. W., On the Schwarz-Christqffel transformation and p-valent functions, Trans. Amer. Math. Soc. 68 (1950), 204–223.Google Scholar
2. Hummel, J. A., The coefficient regions of starlike functions, Pacific J. Math. 7 (1957), 1381–1389.Google Scholar
3. Hummel, J. A., Multivalent starlike functions, J. Analyse Math. 18 (1967), 133–160.Google Scholar
4. Hummel, J. A., Problem 21, Classical function theory problems, Bull. Amer. Math. Soc. 68 (1962), 21–24.Google Scholar