Certain Conditions for Starlikeness of Analytic Functions of Koebe Type
For 𝛼≥0, 𝜆>0, we consider the 𝑀(𝛼,𝜆)𝑏 of normalized analytic 𝛼−𝜆 convex functions defined in the open unit disc 𝕌. In this paper, we investigate the class 𝑀(𝛼,𝜆)𝑏, that is, Re{(𝑧𝑓𝑏(𝑧)/𝑓𝑏(𝑧))[1−𝛼+𝛼(1−𝜆)(𝑧𝑓𝑏(𝑧)/𝑓𝑏(𝑧))+𝛼𝜆(1+(𝑧𝑓𝑏(𝑧)/𝑓𝑏(𝑧)))]}>0, with 𝑓𝑏 is Koebe type, that is, 𝑓𝑏(𝑧)∶=𝑧/(1−𝑧𝑛)...
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2011-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2011/679704 |
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| author | Saibah Siregar Maslina Darus |
| author_facet | Saibah Siregar Maslina Darus |
| author_sort | Saibah Siregar |
| collection | DOAJ |
| description | For 𝛼≥0, 𝜆>0, we consider the 𝑀(𝛼,𝜆)𝑏 of normalized analytic 𝛼−𝜆 convex functions defined in the open unit disc 𝕌. In this paper, we investigate the class 𝑀(𝛼,𝜆)𝑏, that is, Re{(𝑧𝑓𝑏(𝑧)/𝑓𝑏(𝑧))[1−𝛼+𝛼(1−𝜆)(𝑧𝑓𝑏(𝑧)/𝑓𝑏(𝑧))+𝛼𝜆(1+(𝑧𝑓𝑏(𝑧)/𝑓𝑏(𝑧)))]}>0, with 𝑓𝑏 is Koebe type, that is, 𝑓𝑏(𝑧)∶=𝑧/(1−𝑧𝑛)𝑏. The subordination result for the aforementioned class will be given. Further, by making use of Jack's Lemma as well as several differential and other inequalities, the authors derived sufficient conditions for starlikeness of the class 𝑀(𝛼,𝜆)𝑏 of 𝑛-fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in the earlier works are also indicated. |
| format | Article |
| id | doaj-art-e0ce3ee9d6eb46eaa2b88dccf070bbfc |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
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| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-e0ce3ee9d6eb46eaa2b88dccf070bbfc2025-08-20T02:21:30ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252011-01-01201110.1155/2011/679704679704Certain Conditions for Starlikeness of Analytic Functions of Koebe TypeSaibah Siregar0Maslina Darus1Faculty of Science and Biotechnology, Universiti Selangor, Bestari Jaya, 45600 Selangor Darul Ehsan, MalaysiaSchool of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, 43600 Selangor Darul Ehsan, MalaysiaFor 𝛼≥0, 𝜆>0, we consider the 𝑀(𝛼,𝜆)𝑏 of normalized analytic 𝛼−𝜆 convex functions defined in the open unit disc 𝕌. In this paper, we investigate the class 𝑀(𝛼,𝜆)𝑏, that is, Re{(𝑧𝑓𝑏(𝑧)/𝑓𝑏(𝑧))[1−𝛼+𝛼(1−𝜆)(𝑧𝑓𝑏(𝑧)/𝑓𝑏(𝑧))+𝛼𝜆(1+(𝑧𝑓𝑏(𝑧)/𝑓𝑏(𝑧)))]}>0, with 𝑓𝑏 is Koebe type, that is, 𝑓𝑏(𝑧)∶=𝑧/(1−𝑧𝑛)𝑏. The subordination result for the aforementioned class will be given. Further, by making use of Jack's Lemma as well as several differential and other inequalities, the authors derived sufficient conditions for starlikeness of the class 𝑀(𝛼,𝜆)𝑏 of 𝑛-fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in the earlier works are also indicated.http://dx.doi.org/10.1155/2011/679704 |
| spellingShingle | Saibah Siregar Maslina Darus Certain Conditions for Starlikeness of Analytic Functions of Koebe Type International Journal of Mathematics and Mathematical Sciences |
| title | Certain Conditions for Starlikeness of Analytic Functions of Koebe Type |
| title_full | Certain Conditions for Starlikeness of Analytic Functions of Koebe Type |
| title_fullStr | Certain Conditions for Starlikeness of Analytic Functions of Koebe Type |
| title_full_unstemmed | Certain Conditions for Starlikeness of Analytic Functions of Koebe Type |
| title_short | Certain Conditions for Starlikeness of Analytic Functions of Koebe Type |
| title_sort | certain conditions for starlikeness of analytic functions of koebe type |
| url | http://dx.doi.org/10.1155/2011/679704 |
| work_keys_str_mv | AT saibahsiregar certainconditionsforstarlikenessofanalyticfunctionsofkoebetype AT maslinadarus certainconditionsforstarlikenessofanalyticfunctionsofkoebetype |