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Geometric properties and sections for certain subclasses of harmonic mappings
Let G H k ( α ; r ) denote the subclasses of normalized harmonic mappings f = h + g ¯ in the unit disk D satisfying the condition Re ( ( 1 - α ) h ( z ) z + α h ′ ( z ) ) > | ( 1 - α ) g ( z ) z + α g ′ ( z ) | in | z | < r , r ∈ ( 0 , 1 ] , where h ′ ( 0 ) = 1 , g ′ ( 0 ) = h ′ ′ ( 0 ) = ⋯ =...
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| Published in: | Monatshefte für Mathematik 2019-10, Vol.190 (2), p.353-387 |
|---|---|
| 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
G
H
k
(
α
;
r
)
denote the subclasses of normalized harmonic mappings
f
=
h
+
g
¯
in the unit disk
D
satisfying the condition
Re
(
(
1
-
α
)
h
(
z
)
z
+
α
h
′
(
z
)
)
>
|
(
1
-
α
)
g
(
z
)
z
+
α
g
′
(
z
)
|
in
|
z
|
<
r
,
r
∈
(
0
,
1
]
, where
h
′
(
0
)
=
1
,
g
′
(
0
)
=
h
′
′
(
0
)
=
⋯
=
h
(
k
)
(
0
)
=
g
(
k
)
(
0
)
=
0
and
α
≥
0
. In this paper, we first provide the sharp coefficient estimates and the sharp growth theorems for harmonic mappings in the class
G
H
k
(
α
;
1
)
. Next, we derive the geometric properties of harmonic mappings in
G
H
1
(
α
;
1
)
. Then we study several properties of the sections of
f
∈
G
H
k
(
α
;
1
)
. Finally, we show that if
f
∈
P
H
0
(
α
)
and
F
∈
G
H
1
(
β
;
1
)
, then the harmonic convolution
f
∗
F
is univalent and close-to-convex harmonic function in the unit disk for
α
∈
[
1
2
,
1
)
,
β
≥
0
. |
|---|---|
| ISSN: | 0026-9255 1436-5081 |
| DOI: | 10.1007/s00605-018-1240-5 |