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 |
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Wiley
1996-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171296001093 |
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| _version_ | 1832560331561369600 |
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| author | Intisar Qumsiyeh Hibschweiler |
| author_facet | Intisar Qumsiyeh Hibschweiler |
| author_sort | Intisar Qumsiyeh Hibschweiler |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-9e888701f4744c65976e80dffcaa3465 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1996-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-9e888701f4744c65976e80dffcaa34652025-02-03T01:27:52ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251996-01-0119478979510.1155/S0161171296001093Univalent functions maximizing Re[f(ζ1)+f(ζ2)]Intisar Qumsiyeh Hibschweiler0Daemen College, 4380 Main Street, Amherst 14226, New York , USAWe 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.http://dx.doi.org/10.1155/S0161171296001093Univalent FunctionsSupport PointsQuadratic Differential. |
| spellingShingle | Intisar Qumsiyeh Hibschweiler Univalent functions maximizing Re[f(ζ1)+f(ζ2)] International Journal of Mathematics and Mathematical Sciences Univalent Functions Support Points Quadratic Differential. |
| title | Univalent functions maximizing
Re[f(ζ1)+f(ζ2)] |
| title_full | Univalent functions maximizing
Re[f(ζ1)+f(ζ2)] |
| title_fullStr | Univalent functions maximizing
Re[f(ζ1)+f(ζ2)] |
| title_full_unstemmed | Univalent functions maximizing
Re[f(ζ1)+f(ζ2)] |
| title_short | Univalent functions maximizing
Re[f(ζ1)+f(ζ2)] |
| title_sort | univalent functions maximizing re f ζ1 f ζ2 |
| topic | Univalent Functions Support Points Quadratic Differential. |
| url | http://dx.doi.org/10.1155/S0161171296001093 |
| work_keys_str_mv | AT intisarqumsiyehhibschweiler univalentfunctionsmaximizingrefz1fz2 |