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Area and length maxima for univalent functions

Published online by Cambridge University Press:  17 April 2009

Shinji Yamashita
Shinji Yamashita
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, Fukasawa, Setagaya, Tokyo 158, Japan
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Let S be the family of functions f(z) = z + a2z2 + … which are analytic and univalent in |z| < 1. We find the value

as a function of r 0 < r < 1. The known lower estimate of

is improved. Relations with the growth theorem are considered and the radius of univalence of f(z)/z is discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 41 , Issue 3 , June 1990 , pp. 435 - 439
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]de Branges, L., ‘A proof of the Bieberbach conjecture’, Acta Math. 154 (1985), 137–152.CrossRefGoogle Scholar
[2]Duren, P.L., ‘An arclength problem for close–to–convex functions’, J. London Math. Soc. 39 (1964), 757–761.Google Scholar
[3]Duren, P.L., Univalent functions (Springer–Verlag, New York, Berlin, Heidelberg, Tokyo, 1983).Google Scholar