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QUASICONFORMAL EXTENDABILITY OF INTEGRAL TRANSFORMS OF NOSHIRO-WARSCHAWSKI FUNCTIONS
Since the nonlinear integral transforms J α [ f ] ( z ) = ∫ 0 z ( f ' ( u ) ) α d u and I α [ f ] ( z ) = ∫ 0 z ( f ( u ) / u ) α d u with a complex number α were introduced, a great number of studies have been dedicated to deriving suffcient conditions for univalence on the unit disk. However,...
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| Published in: | The Rocky Mountain journal of mathematics 2017-01, Vol.47 (1), p.185-204 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | Since the nonlinear integral transforms
J
α
[
f
]
(
z
)
=
∫
0
z
(
f
'
(
u
)
)
α
d
u
and
I
α
[
f
]
(
z
)
=
∫
0
z
(
f
(
u
)
/
u
)
α
d
u
with a complex number α were introduced, a great number of studies have been dedicated to deriving suffcient conditions for univalence on the unit disk. However, little is known about the conditions where Jα[f] or Iα[f] produce a holomorphic univalent function in the unit disk which extends to a quasiconformal map on the complex plane. In this paper, we discuss quasiconformal extendability of the integral transforms Jα[f] and Iα[f] for holomorphic functions which satisfy the Noshiro-Warschawski criterion. Various approaches using pre-Schwarzian derivatives, differential subordination and Loewner theory are applied to this problem. |
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| ISSN: | 0035-7596 1945-3795 |
| DOI: | 10.1216/rmj-2017-47-1-185 |