Loading…
On the Coefficients of Certain Subclasses of Harmonic Univalent Mappings with Nonzero Pole
Let Co ( p ), p ∈ ( 0 , 1 ] be the class of all meromorphic univalent functions φ defined in the open unit disc D with normalizations φ ( 0 ) = 0 = φ ′ ( 0 ) - 1 and having simple pole at z = p ∈ ( 0 , 1 ] such that the complement of φ ( D ) is a convex domain. The class Co ( p ) is called the class...
Saved in:
| Published in: | Boletim da Sociedade Brasileira de Matemática 2021-12, Vol.52 (4), p.1041-1053 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | Let
Co
(
p
),
p
∈
(
0
,
1
]
be the class of all meromorphic univalent functions
φ
defined in the open unit disc
D
with normalizations
φ
(
0
)
=
0
=
φ
′
(
0
)
-
1
and having simple pole at
z
=
p
∈
(
0
,
1
]
such that the complement of
φ
(
D
)
is a convex domain. The class
Co
(
p
) is called the class of concave univalent functions. Let
S
H
0
(
p
)
be the class of all sense preserving univalent harmonic mappings
f
defined on
D
having simple pole at
z
=
p
∈
(
0
,
1
)
with the normalizations
f
(
0
)
=
f
z
(
0
)
-
1
=
0
and
f
z
¯
(
0
)
=
0
. We first derive the exact regions of variability for the second Taylor coefficients of
h
where
f
=
h
+
g
¯
∈
S
H
0
(
p
)
with
h
-
g
∈
C
o
(
p
)
. Next we consider the class
S
H
0
(
1
)
of all sense preserving univalent harmonic mappings
f
in
D
having simple pole at
z
=
1
with the same normalizations as above. We derive exact regions of variability for the coefficients of
h
where
f
=
h
+
g
¯
∈
S
H
0
(
1
)
satisfying
h
-
e
2
i
θ
g
∈
C
o
(
1
)
with dilatation
g
′
(
z
)
/
h
′
(
z
)
=
e
-
2
i
θ
z
, for some
θ
,
0
≤
θ
<
π
. |
|---|---|
| ISSN: | 1678-7544 1678-7714 |
| DOI: | 10.1007/s00574-021-00244-x |