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On Convex Univalent Functions

Published online by Cambridge University Press:  20 November 2018

T. Basgoze,
J. L. Frank and
F. R. Keogh
T. Basgoze
Affiliation:
Middle East Technical University, Ankara, Turkey
J. L. Frank
Affiliation:
University of Kentucky, Lexington, Kentucky
F. R. Keogh
Affiliation:
University of Kentucky, Lexington, Kentucky
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In what follows, we suppose that ƒ(z) = Σ0 ∞anzn is regular for |z| < 1. Let

and

Then (see, for example, [6, pp. 235-236]), for 0 ≦ r < ρ < 1, we have:

The following results are well known.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970, pp. 123 - 127
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Clunie, J. and Keogh, F. R., On starlike and convex schlicht functions, J. London Math. Soc. 85 (1960), 229–233.Google Scholar
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3. Keogh, F. R., A strengthened form of the ¼theorem for starlike univalent functions (to appear in the A. J. Maclntyre Memorial Volume, Ohio Univ. Press).Google Scholar
4. Nehari, Z., Conformai mapping (McGraw-Hill, New York, 1952).Google Scholar
5. G., Pölya and I. J., Schoenberg, Remarks on de la Vallée Poussin means and convex conformai maps of the circle, Pacific J. Math. 8 (1958), 295–334.Google Scholar
6. Titchmarsh, E. C., The theory of functions, 2nd ed. (Oxford Univ. Press, London, 1939).Google Scholar