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Entire Functions with Some Derivatives Univalent

  • S. M. Shah (a1) and S. Y. Trimble (a2)

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This paper is a continuation of the author's previous work, [6; 7], on the relationship between the radius of convergence of a power series and the number of derivatives of the power series which are univalent in a given disc.

In particular, let D be the open disc centered at 0, and let f be regular there. Suppose that is a strictly-increasing sequence of positive integers such that each f(n p ) is univalent in D. Let R be the radius of convergence of the power series, centered at 0, that represents f. In [7], we investigated the connection between R and . We showed that, in general

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References

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1. FitzGerald, C. H., Exponentiation of certain quadratic inequalities for Schlicht functions, Bull. Amer. Math. Soc. 78 (1972), 209210.
2. Hayman, W. K., Multivalent functions (Cambridge University Press, Cambridge, 1958).
3. Hobson, E. W., The theory of functions of a real variable and the theory of Fourier's series, vol. II (Dover Publications, New York, 1957).
4. Mitrinovic, D. S., Analytic inequalities (Springer-Verlag, New York, 1970).
5. Nehari, Z., Conformai mapping (McGraw-Hill, New York, 1952).
6. Shah, S. M. and Trimble, S. Y., Univalent functions with univalent derivatives, Bull. Amer. Math. Soc. 75 (1969), 153157.
7. Shah, S. M. and Trimble, S. Y., Univalent functions with univalent derivatives, III, J. Math. Mech. 19 (1969), 451460.
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Entire Functions with Some Derivatives Univalent

  • S. M. Shah (a1) and S. Y. Trimble (a2)

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