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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
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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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |