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## Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

2010 | 64 | 1 |
Tytuł artykułu

### Inclusion properties of certain subclasses of analytic functions defined by generalized Salagean operator

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Let $$A$$ denote the class of analytic functions with the normalization $$f(0)=f^{\prime }(0)-1=0$$ in the open unit disc $$U=\{z:\left\vert z\right\vert <1\}$$.  Set $f_{\lambda }^{n}(z)=z+\sum_{k=2}^{\infty }[1+\lambda (k-1)]^{n}z^{k}\quad(n\in N_{0};\ \lambda \geq 0;\ z\in U),$ and define $$f_{\lambda ,\mu }^{n}$$ in terms of the Hadamard product $f_{\lambda }^{n}(z)\ast f_{\lambda ,\mu }^{n}=\frac{z}{(1-z)^{\mu }}\quad (\mu >0;\ z\in U).$ In this paper, we introduce several subclasses of analytic functions defined by means of the operator $$I_{\lambda ,\mu }^{n}:A\longrightarrow A$$, given by $I_{\lambda ,\mu }^{n}f(z)=f_{\lambda ,\mu }^{n}(z)\ast f(z)\quad (f\in A;\ n\in N_{0;}\ \lambda \geq 0;\ \mu >0).$Inclusion properties of these classes and the classes involving the generalized Libera integral operator are also considered.
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2010
online
2016-07-29
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Bibliografia
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• Bernardi, S. D., Convex and starlike univalent functions, Trans. Amer. Math. Soc. 35 (1969), 429-446.
• Choi, J. H., Saigo, M. and Srivastava, H. M., Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl. 276 (2002), 432-445.
• Eenigenburg, P., Miller, S. S., Mocanu, P. T. and Reade, M. O., On a Briot–Bouquet differential subordination, General inequalities, 3 (Oberwolfach, 1981), 339-348, Internat. Schriftenreihe Numer. Math., 64, Birkhauser, Basel, 1983.
• Kim, Y. C., Choi, J. H. and Sugawa, T., Coefficient bounds and convolution properties for certain classes of close-to-convex functions, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 95-98.
• Libera, R. J., Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755-758.
• Ma, W. C., Minda, D., An internal geometric characterization of strongly starlike functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 45 (1991), 89-97.
• Miller, S. S., Mocanu, P. T., Differential subordinations and univalent functions, Michigan Math. J. 28 (1981), 157-171.
• Owa, S., Srivastava, H. M., Some applications of the generalized Libera operator, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), 125-128.
• Salagean, G. S., Subclasses of univalent functions, Complex analysis - fifth
• Romanian-Finnish seminar, Part 1 (Bucharest, 1981), 362-372, Lecture Notes in Math., 1013, Springer, Berlin, 1983.
• Srivastava, H. M., Owa, S. (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, 1992.
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