International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 4, Pages 789-795
doi:10.1155/S0161171296001093
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 Intisar Qumsiyeh Hibschweiler. 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.
Abstract
We study the problem maxh∈Sℜ[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.