{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:41:53Z","timestamp":1740134513017,"version":"3.37.3"},"reference-count":18,"publisher":"Wiley","license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2015]]},"abstract":"<jats:p>Let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi mathvariant=\"double-struck\">D<\/mml:mi><\/mml:mrow><\/mml:math>denote the open unit disk and let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi mathvariant=\"script\">S<\/mml:mi><\/mml:mrow><\/mml:math>denote the class of normalized univalent functions which are analytic in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi mathvariant=\"double-struck\">D<\/mml:mi><\/mml:mrow><\/mml:math>. Let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mtext>C<\/mml:mtext><mml:mtext>o<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b1<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>be the class of concave functions<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>f<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi mathvariant=\"script\">S<\/mml:mi><\/mml:math>, which have the condition that the opening angle of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi mathvariant=\"double-struck\">D<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>at infinity is less than or equal to<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>\u03c0<\/mml:mi><mml:mi>\u03b1<\/mml:mi><\/mml:math>,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mi>\u03b1<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>1,2<\/mml:mn><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:math>. In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mtext>C<\/mml:mtext><mml:mtext>o<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b1<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. And we define a class<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mtext>C<\/mml:mtext><mml:mtext>o<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b1<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>B<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>\u2264<\/mml:mo><mml:mi>B<\/mml:mi><mml:mo>&lt;<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo>\u2264<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, which is a subclass of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mtext>C<\/mml:mtext><mml:mtext>o<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b1<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>and we find the set of variabilities for the functional<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>-<\/mml:mo><mml:mo stretchy=\"false\">|<\/mml:mo><mml:mi>z<\/mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u2033<\/mml:mo><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>\/<\/mml:mo><mml:msup><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>f<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mtext>C<\/mml:mtext><mml:mtext>o<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b1<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>B<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. This gives sharp upper and lower estimates for the pre-Schwarzian norm of functions in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mtext>C<\/mml:mtext><mml:mtext>o<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b1<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>B<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. We also give a characterization for functions in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mtext>C<\/mml:mtext><mml:mtext>o<\/mml:mtext><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>\u03b1<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>B<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>in terms of Hadamard product.<\/jats:p>","DOI":"10.1155\/2015\/814805","type":"journal-article","created":{"date-parts":[[2015,4,15]],"date-time":"2015-04-15T17:01:51Z","timestamp":1429117311000},"page":"1-6","source":"Crossref","is-referenced-by-count":1,"title":["The Pre-Schwarzian Norm Estimate for Analytic Concave Functions"],"prefix":"10.1155","volume":"2015","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6741-9752","authenticated-orcid":true,"given":"Young Jae","family":"Sim","sequence":"first","affiliation":[{"name":"Department of Mathematics, Kyungsung University, Busan 608-736, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Oh Sang","family":"Kwon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kyungsung University, Busan 608-736, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","first-page":"3","volume":"17","year":"2005","journal-title":"Lobachevskii Journal of Mathematics"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1007\/s00605-009-0146-7"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.4171\/cmh\/74"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1080\/02781070290016223"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2010.07.054"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1080\/17476930601063693"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.2989\/16073600509486117"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1137\/0515057"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1090\/s0002-9939-1961-0143950-1"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(98)00090-9"},{"key":"11","first-page":"1","volume":"52","year":"1998","journal-title":"Annales Universitatis Mariae Curie-Sk\u0142odowska Sectio A"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1993.1044"},{"issue":"1","key":"13","first-page":"1","volume":"20","year":"2004","journal-title":"Tamsui Oxford Journal of Mathematical Sciences"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1155\/s0161171299227652"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(97)00221-5"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1216\/rmjm\/1008959684"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(86)90329-X"},{"year":"2000","key":"18"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2015\/814805.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2015\/814805.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2015\/814805.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,27]],"date-time":"2016-07-27T15:10:53Z","timestamp":1469632253000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2015\/814805\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"references-count":18,"alternative-id":["814805","814805"],"URL":"https:\/\/doi.org\/10.1155\/2015\/814805","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2015]]}}}