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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 |
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| cites | cdi_FETCH-LOGICAL-c316t-fdd4b06ce31c294c392e8417587f90958acfd8df6842419c8ef50b8de9818fa63 |
| container_end_page | 387 |
| container_issue | 2 |
| container_start_page | 353 |
| container_title | Monatshefte für Mathematik |
| container_volume | 190 |
| creator | Liu, Ming-Sheng Yang, Li-Mei |
| description | 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
. |
| doi_str_mv | 10.1007/s00605-018-1240-5 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2291465371</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2291465371</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-fdd4b06ce31c294c392e8417587f90958acfd8df6842419c8ef50b8de9818fa63</originalsourceid><addsrcrecordid>eNp1kD1PwzAQhi0EEqXwA9giMRvuHNuxR1TxJRWxwGy5jl1SNXGw04F_j6sgMTHdcM_z3ukl5BrhFgGauwwgQVBARZFxoOKELJDXkgpQeEoWAExSzYQ4Jxc57wAAa6kX5PXJx95PqXPVmOLo09T5XNmhrbJ3UxeHXIWYKlcWthuqfNi4vc25MDFUnzb1cShqb8exG7b5kpwFu8_-6ncuycfjw_vqma7fnl5W92vqapQTDW3LNyCdr9ExzV2tmVccG6GaoEELZV1oVRuk4oyjdsoHARvVeq1QBSvrJbmZc8vPXwefJ7OLhzSUk4YxjVyKusFC4Uy5FHNOPpgxdb1N3wbBHFszc2umtGaOrRlRHDY7ubDD1qe_5P-lHxlXb-U</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2291465371</pqid></control><display><type>article</type><title>Geometric properties and sections for certain subclasses of harmonic mappings</title><source>Springer Link</source><creator>Liu, Ming-Sheng ; Yang, Li-Mei</creator><creatorcontrib>Liu, Ming-Sheng ; Yang, Li-Mei</creatorcontrib><description>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
.</description><identifier>ISSN: 0026-9255</identifier><identifier>EISSN: 1436-5081</identifier><identifier>DOI: 10.1007/s00605-018-1240-5</identifier><language>eng</language><publisher>Vienna: Springer Vienna</publisher><subject>Convolution ; Harmonic functions ; Mathematics ; Mathematics and Statistics ; Properties (attributes) ; Theorems</subject><ispartof>Monatshefte für Mathematik, 2019-10, Vol.190 (2), p.353-387</ispartof><rights>Springer-Verlag GmbH Austria, ein Teil von Springer Nature 2018</rights><rights>Copyright Springer Nature B.V. 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-fdd4b06ce31c294c392e8417587f90958acfd8df6842419c8ef50b8de9818fa63</citedby><cites>FETCH-LOGICAL-c316t-fdd4b06ce31c294c392e8417587f90958acfd8df6842419c8ef50b8de9818fa63</cites><orcidid>0000-0002-2644-6997</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27922,27923</link.rule.ids></links><search><creatorcontrib>Liu, Ming-Sheng</creatorcontrib><creatorcontrib>Yang, Li-Mei</creatorcontrib><title>Geometric properties and sections for certain subclasses of harmonic mappings</title><title>Monatshefte für Mathematik</title><addtitle>Monatsh Math</addtitle><description>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
.</description><subject>Convolution</subject><subject>Harmonic functions</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Properties (attributes)</subject><subject>Theorems</subject><issn>0026-9255</issn><issn>1436-5081</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp1kD1PwzAQhi0EEqXwA9giMRvuHNuxR1TxJRWxwGy5jl1SNXGw04F_j6sgMTHdcM_z3ukl5BrhFgGauwwgQVBARZFxoOKELJDXkgpQeEoWAExSzYQ4Jxc57wAAa6kX5PXJx95PqXPVmOLo09T5XNmhrbJ3UxeHXIWYKlcWthuqfNi4vc25MDFUnzb1cShqb8exG7b5kpwFu8_-6ncuycfjw_vqma7fnl5W92vqapQTDW3LNyCdr9ExzV2tmVccG6GaoEELZV1oVRuk4oyjdsoHARvVeq1QBSvrJbmZc8vPXwefJ7OLhzSUk4YxjVyKusFC4Uy5FHNOPpgxdb1N3wbBHFszc2umtGaOrRlRHDY7ubDD1qe_5P-lHxlXb-U</recordid><startdate>20191001</startdate><enddate>20191001</enddate><creator>Liu, Ming-Sheng</creator><creator>Yang, Li-Mei</creator><general>Springer Vienna</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-2644-6997</orcidid></search><sort><creationdate>20191001</creationdate><title>Geometric properties and sections for certain subclasses of harmonic mappings</title><author>Liu, Ming-Sheng ; Yang, Li-Mei</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-fdd4b06ce31c294c392e8417587f90958acfd8df6842419c8ef50b8de9818fa63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Convolution</topic><topic>Harmonic functions</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Properties (attributes)</topic><topic>Theorems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Liu, Ming-Sheng</creatorcontrib><creatorcontrib>Yang, Li-Mei</creatorcontrib><collection>CrossRef</collection><jtitle>Monatshefte für Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Liu, Ming-Sheng</au><au>Yang, Li-Mei</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Geometric properties and sections for certain subclasses of harmonic mappings</atitle><jtitle>Monatshefte für Mathematik</jtitle><stitle>Monatsh Math</stitle><date>2019-10-01</date><risdate>2019</risdate><volume>190</volume><issue>2</issue><spage>353</spage><epage>387</epage><pages>353-387</pages><issn>0026-9255</issn><eissn>1436-5081</eissn><abstract>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
.</abstract><cop>Vienna</cop><pub>Springer Vienna</pub><doi>10.1007/s00605-018-1240-5</doi><tpages>35</tpages><orcidid>https://orcid.org/0000-0002-2644-6997</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0026-9255 |
| ispartof | Monatshefte für Mathematik, 2019-10, Vol.190 (2), p.353-387 |
| issn | 0026-9255 1436-5081 |
| language | eng |
| recordid | cdi_proquest_journals_2291465371 |
| source | Springer Link |
| subjects | Convolution Harmonic functions Mathematics Mathematics and Statistics Properties (attributes) Theorems |
| title | Geometric properties and sections for certain subclasses of harmonic mappings |
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