{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:42:00Z","timestamp":1740134520396,"version":"3.37.3"},"reference-count":7,"publisher":"Wiley","license":[{"start":{"date-parts":[[2018,1,1]],"date-time":"2018-01-01T00:00:00Z","timestamp":1514764800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2018]]},"abstract":"<jats:p>For <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>j<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">1,2<\/mml:mn><\/mml:math> and for positive integers <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math>, we consider classes of harmonic functions <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>j<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>j<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mover accent=\"false\"><mml:mrow><mml:msub><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>j<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mo>\u00af<\/mml:mo><\/mml:mover><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:msub><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><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>z<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:msub><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:msubsup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msubsup><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>z<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:msubsup><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> or <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:msubsup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msubsup><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>z<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:msubsup><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:msubsup><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msubsup><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>z<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msup><mml:msubsup><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u2032<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>z<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>, and we prove that their convolution <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>\u204e<\/mml:mo><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>\u204e<\/mml:mo><mml:msub><mml:mrow><mml:mi>h<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mover accent=\"false\"><mml:mrow><mml:msub><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>\u204e<\/mml:mo><mml:msub><mml:mrow><mml:mi>g<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mo>\u00af<\/mml:mo><\/mml:mover><\/mml:math> is locally one-to-one, sense-preserving, and close-to-convex harmonic in <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>z<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>&lt;<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2018\/3808513","type":"journal-article","created":{"date-parts":[[2018,5,2]],"date-time":"2018-05-02T19:35:36Z","timestamp":1525289736000},"page":"1-4","source":"Crossref","is-referenced-by-count":0,"title":["Close-to-Convexity of Convolutions of Classes of Harmonic Functions"],"prefix":"10.1155","volume":"2018","author":[{"given":"Raj Kumar","family":"Garg","sequence":"first","affiliation":[{"name":"DAV University, Jalandhar, Punjab 144012, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5231-3530","authenticated-orcid":true,"given":"Jay M.","family":"Jahangiri","sequence":"additional","affiliation":[{"name":"Mathematical Sciences, Kent State University, Burton, OH 44021-9500, USA"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1936-06397-4"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.5186\/aasfm.1984.0905"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511546600"},{"volume":"259","year":"1983","key":"4"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02566116"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1155\/2017\/4015268"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1090\/s0002-9947-1947-0019114-6"}],"container-title":["International Journal of Mathematics and Mathematical 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