{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,21]],"date-time":"2026-06-21T17:32:15Z","timestamp":1782063135308,"version":"3.54.5"},"reference-count":14,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2007,1,1]],"date-time":"2007-01-01T00:00:00Z","timestamp":1167609600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"Intensification of Research in Priority Areas","award":["09-02-05-00020 EAR"],"award-info":[{"award-number":["09-02-05-00020 EAR"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2007]]},"abstract":"<jats:p>Let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E1\"><mml:mrow><mml:mi>A<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>B<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>D<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>E<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mrow><mml:mo>[<\/mml:mo><mml:mrow><mml:mo>\u2212<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mo>]<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>and let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E2\"><mml:mrow><mml:mi>p<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>be an analytic function defined on the open unit disk,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E3\"><mml:mrow><mml:mi>p<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:math>. Conditions on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E4\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:math>,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E5\"><mml:mrow><mml:mi>B<\/mml:mi><\/mml:mrow><\/mml:math>,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E6\"><mml:mrow><mml:mi>D<\/mml:mi><\/mml:mrow><\/mml:math>, and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E7\"><mml:mi>E<\/mml:mi><\/mml:math>are determined so that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E8\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>\u03b2<\/mml:mi><mml:mi>z<\/mml:mi><mml:mi>p<\/mml:mi><mml:mo>'<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math>being subordinated to<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E9\"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>D<\/mml:mi><mml:mi>z<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>E<\/mml:mi><mml:mi>z<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:math>implies that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E10\"><mml:mrow><mml:mi>p<\/mml:mi><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math>is subordinated to<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E11\"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>A<\/mml:mi><mml:mi>z<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mo>\/<\/mml:mo><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>B<\/mml:mi><mml:mi>z<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:math>. Similar results are obtained by considering the expressions<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E12\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>\u03b2<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mi>p<\/mml:mi><mml:mo>'<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo>)<\/mml:mo><mml:mo>\/<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo>)<\/mml:mo><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"E13\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>\u03b2<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mi>p<\/mml:mi><mml:mo>'<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo>)<\/mml:mo><mml:mo>\/<\/mml:mo><mml:msup><mml:mi>p<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><mml:mo>(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo>)<\/mml:mo><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:math>. These results are then applied to obtain sufficient conditions for analytic functions to be Janowski starlike.<\/jats:p>","DOI":"10.1155\/2007\/62925","type":"journal-article","created":{"date-parts":[[2007,8,26]],"date-time":"2007-08-26T06:24:29Z","timestamp":1188109469000},"page":"1-7","source":"Crossref","is-referenced-by-count":28,"title":["Sufficient Conditions for Janowski Starlikeness"],"prefix":"10.1155","volume":"2007","author":[{"given":"Rosihan M.","family":"Ali","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"V.","family":"Ravichandran","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"N.","family":"Seenivasagan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"98","reference":[{"key":"1","doi-asserted-by":"crossref","first-page":"297","DOI":"10.4064\/ap-28-3-297-326","volume":"28","year":"1973","journal-title":"Annales Polonici Mathematici"},{"issue":"1-2","key":"2","first-page":"9","volume":"54","year":"2002","journal-title":"Matematichki Vesnik"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1155\/S0161171299220753"},{"issue":"24","key":"4","first-page":"59","year":"2000","journal-title":"Zeszyty Naukowe Politechniki Rzeszowskiej. 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