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On a maximal outer area problem for a class of meromorphic univalent fuctions

Published online by Cambridge University Press:  17 April 2009

Stephen M. Zemyan
Stephen M. Zemyan
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802.
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Abstract

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For 0 < p < 1, let Sp denote the class of functions f (z) which are meromorphic and univalent in the unit disk U, with the normalisations f (0) = 0, f′(0) = 1 and f (p) = ∞, and let Sp (a) denote subclass of Sp consisting of those functions in Sp whose residue at the pole in equal to a. In this paper, we determine, for values of the residue a in a certain disk Δp, the greatest possible outer area over all functions in the class Sp (a). We also determine additional information concerning extremal function if the reside a dose not lie in Δp.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 3 , December 1986 , pp. 433 - 445
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Kober, H., Dictionary of Conformal Representations, Dover Publications, New York (1957).Google Scholar
[2]Komatu, Y., “Note on the theory of conformal representation by meromorphic functions I and II”, Proc. Japan Acad. 21 (1945), 269–277; 278–284.CrossRefGoogle Scholar
[3]Ladegast, K., “Beiträge zur Theorie der schlichten Funktionen”, Math. Z., 58 (1953), 115–159.CrossRefGoogle Scholar
[4]Lewandowski, Z. and Złotkiewicz, E., “Variational formulae for function meromorphic and univalent in the unit disc”, Ann. Univ. Mariae Curie - Skłodowska 17 (1963), 47–53.Google Scholar
[5]Lewandowski, Z. and Złotkiewicz, E., “Variational formulae for functions meromorphic and univalent in the unit disc”, Bull. Acad. Polon, Sci., Math., XII (1964), No.5, 253–254.Google Scholar
[6]Lewandowski, Z., Libera, R. J. and Złotkiewicz, E., “Mapping properites of a class of univalent functions with pre-assigned zero and pole”, Ann. polon. math., XL (1983), 283–289.CrossRefGoogle Scholar
[7]Zemyan, S., “A minimal outer area problem in conformal mapping”, J. Analyse Math., 39 (1981), 11–23.CrossRefGoogle Scholar
[8]Zemyan, S., “The range of the residue functional for the class Sp” Michigan Math. J., 31 (1984), 73–77.CrossRefGoogle Scholar