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International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 789-795

Univalent functions maximizing Re[f(ζ1)+f(ζ2)]

Daemen College, 4380 Main Street, Amherst 14226, New York , USA

Received 11 April 1994; Revised 28 September 1995

Copyright © 1996 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the problem maxhS[h(z1)+h(z2)] with z1,z2 in Δ. We show that no rotation of the Koebe function is a solution for this problem except possibly its real rotation, and only when z1=z¯2 or z1,z2 are both real, and are in a neighborhood of the x-axis. We prove that if the omitted set of the extremal function f is part of a straight line that passes through f(z1) or f(z2) then f is the Koebe function or its real rotation. We also show the existence of solutions that are not unique and are different from the Koebe function or its real rotation. The situation where the extremal value is equal to zero can occur and it is proved, in this case, that the Koebe function is a solution if and only if z1 and z2 are both real numbers and z1z2<0.