The spherical metric and univalent harmonic mappings
Let f = h + g ¯ be a harmonic univalent map in the unit disk D , where h and g are analytic. This paper finds an improved estimate for the second coefficient of h . Indeed, this estimate is the first qualitative improvement since the appearance of the papers by Clunie and Sheil-Small (Ann Acad Sci F...
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| Veröffentlicht in: | Monatshefte für Mathematik 2019-04, Vol.188 (4), p.703-716 |
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| creator | Abu Muhanna, Yusuf Ali, Rosihan M. Ponnusamy, Saminathan |
| description | Let
f
=
h
+
g
¯
be a harmonic univalent map in the unit disk
D
, where
h
and
g
are analytic. This paper finds an improved estimate for the second coefficient of
h
. Indeed, this estimate is the first qualitative improvement since the appearance of the papers by Clunie and Sheil-Small (Ann Acad Sci Fenn Ser A I 9:3–25,
1984
), and by Sheil-Small (J Lond Math Soc 42:237–248,
1990
). When the sup-norm of the dilatation is less than 1, it is also shown that the spherical area of the covering surface of
h
is dominated by the spherical area of the covering surface of
f
. |
| doi_str_mv | 10.1007/s00605-018-1160-4 |
| format | Article |
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f
=
h
+
g
¯
be a harmonic univalent map in the unit disk
D
, where
h
and
g
are analytic. This paper finds an improved estimate for the second coefficient of
h
. Indeed, this estimate is the first qualitative improvement since the appearance of the papers by Clunie and Sheil-Small (Ann Acad Sci Fenn Ser A I 9:3–25,
1984
), and by Sheil-Small (J Lond Math Soc 42:237–248,
1990
). When the sup-norm of the dilatation is less than 1, it is also shown that the spherical area of the covering surface of
h
is dominated by the spherical area of the covering surface of
f
.</description><identifier>ISSN: 0026-9255</identifier><identifier>EISSN: 1436-5081</identifier><identifier>DOI: 10.1007/s00605-018-1160-4</identifier><language>eng</language><publisher>Vienna: Springer Vienna</publisher><subject>Mathematics ; Mathematics and Statistics ; Spherical harmonics ; Stretching</subject><ispartof>Monatshefte für Mathematik, 2019-04, Vol.188 (4), p.703-716</ispartof><rights>Springer-Verlag GmbH Austria, part of Springer Nature 2018</rights><rights>Copyright Springer Nature B.V. 2019</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-298ee864eedd027236d843d8e0e7933bfb9d62821aa24802908ed1dbff898ef63</citedby><cites>FETCH-LOGICAL-c316t-298ee864eedd027236d843d8e0e7933bfb9d62821aa24802908ed1dbff898ef63</cites><orcidid>0000-0002-3699-2713</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00605-018-1160-4$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00605-018-1160-4$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Abu Muhanna, Yusuf</creatorcontrib><creatorcontrib>Ali, Rosihan M.</creatorcontrib><creatorcontrib>Ponnusamy, Saminathan</creatorcontrib><title>The spherical metric and univalent harmonic mappings</title><title>Monatshefte für Mathematik</title><addtitle>Monatsh Math</addtitle><description>Let
f
=
h
+
g
¯
be a harmonic univalent map in the unit disk
D
, where
h
and
g
are analytic. This paper finds an improved estimate for the second coefficient of
h
. Indeed, this estimate is the first qualitative improvement since the appearance of the papers by Clunie and Sheil-Small (Ann Acad Sci Fenn Ser A I 9:3–25,
1984
), and by Sheil-Small (J Lond Math Soc 42:237–248,
1990
). When the sup-norm of the dilatation is less than 1, it is also shown that the spherical area of the covering surface of
h
is dominated by the spherical area of the covering surface of
f
.</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Spherical harmonics</subject><subject>Stretching</subject><issn>0026-9255</issn><issn>1436-5081</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp1kE1LAzEQhoMoWKs_wNuC5-hMks0mRyl-QcFLPYe0mW23dD9MtoL_3pQVPHmaYXifd-Bh7BbhHgGqhwSgoeSAhiNq4OqMzVBJzUsweM5mAEJzK8rykl2ltAcAlNrOmFrtqEjDjmKz8YeipTEvhe9CceyaL3-gbix2PrZ9l8-tH4am26ZrdlH7Q6Kb3zlnH89Pq8UrX76_vC0el3wjUY9cWENktCIKAUQlpA5GyWAIqLJSruu1DVoYgd4LZUBYMBQwrOvaZLLWcs7upt4h9p9HSqPb98fY5ZdOoDVaigpFTuGU2sQ-pUi1G2LT-vjtENxJjpvkuCzHneQ4lRkxMSlnuy3Fv-b_oR_l0WYK</recordid><startdate>20190411</startdate><enddate>20190411</enddate><creator>Abu Muhanna, Yusuf</creator><creator>Ali, Rosihan M.</creator><creator>Ponnusamy, Saminathan</creator><general>Springer Vienna</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-3699-2713</orcidid></search><sort><creationdate>20190411</creationdate><title>The spherical metric and univalent harmonic mappings</title><author>Abu Muhanna, Yusuf ; Ali, Rosihan M. ; Ponnusamy, Saminathan</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-298ee864eedd027236d843d8e0e7933bfb9d62821aa24802908ed1dbff898ef63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Spherical harmonics</topic><topic>Stretching</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Abu Muhanna, Yusuf</creatorcontrib><creatorcontrib>Ali, Rosihan M.</creatorcontrib><creatorcontrib>Ponnusamy, Saminathan</creatorcontrib><collection>CrossRef</collection><jtitle>Monatshefte für Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Abu Muhanna, Yusuf</au><au>Ali, Rosihan M.</au><au>Ponnusamy, Saminathan</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The spherical metric and univalent harmonic mappings</atitle><jtitle>Monatshefte für Mathematik</jtitle><stitle>Monatsh Math</stitle><date>2019-04-11</date><risdate>2019</risdate><volume>188</volume><issue>4</issue><spage>703</spage><epage>716</epage><pages>703-716</pages><issn>0026-9255</issn><eissn>1436-5081</eissn><abstract>Let
f
=
h
+
g
¯
be a harmonic univalent map in the unit disk
D
, where
h
and
g
are analytic. This paper finds an improved estimate for the second coefficient of
h
. Indeed, this estimate is the first qualitative improvement since the appearance of the papers by Clunie and Sheil-Small (Ann Acad Sci Fenn Ser A I 9:3–25,
1984
), and by Sheil-Small (J Lond Math Soc 42:237–248,
1990
). When the sup-norm of the dilatation is less than 1, it is also shown that the spherical area of the covering surface of
h
is dominated by the spherical area of the covering surface of
f
.</abstract><cop>Vienna</cop><pub>Springer Vienna</pub><doi>10.1007/s00605-018-1160-4</doi><tpages>14</tpages><orcidid>https://orcid.org/0000-0002-3699-2713</orcidid></addata></record> |
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| ispartof | Monatshefte für Mathematik, 2019-04, Vol.188 (4), p.703-716 |
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| language | eng |
| recordid | cdi_proquest_journals_2198632712 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Mathematics Mathematics and Statistics Spherical harmonics Stretching |
| title | The spherical metric and univalent harmonic mappings |
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