Univalent functions maximizing Re[f(ζ1)+f(ζ2)]
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 e...
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
1996-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171296001093 |
| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |