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On a subclass of harmonic close-to-convex mappings
Let H denote the class of harmonic functions f defined in D : = { z ∈ C : | z | < 1 } , and normalized by f ( 0 ) = 0 = f z ( 0 ) - 1 . In this paper, for α ≥ 0 , we consider the subclass W H 0 ( α ) of H , defined by W H 0 ( α ) : = f = h + g ¯ ∈ H : Re ( h ′ ( z ) + α z h ′ ′ ( z ) ) > | g ′...
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| Published in: | Monatshefte für Mathematik 2019-02, Vol.188 (2), p.247-267 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | Let
H
denote the class of harmonic functions
f
defined in
D
:
=
{
z
∈
C
:
|
z
|
<
1
}
, and normalized by
f
(
0
)
=
0
=
f
z
(
0
)
-
1
. In this paper, for
α
≥
0
, we consider the subclass
W
H
0
(
α
)
of
H
, defined by
W
H
0
(
α
)
:
=
f
=
h
+
g
¯
∈
H
:
Re
(
h
′
(
z
)
+
α
z
h
′
′
(
z
)
)
>
|
g
′
(
z
)
+
α
z
g
′
′
(
z
)
|
,
z
∈
D
.
For
f
∈
W
H
0
(
α
)
, we prove the Clunie–Sheil-Small coefficient conjecture, and give some growth, convolution, and convex combination theorems. We also determine the value of
r
so that the partial sums of functions in
W
H
0
(
α
)
are close-to-convex in
|
z
|
<
r
. |
|---|---|
| ISSN: | 0026-9255 1436-5081 |
| DOI: | 10.1007/s00605-017-1138-7 |