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

M. K. Aouf, A. Shamandy, A. O. Mostafa, S. M. Madian


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.


Analytic; Hadamard product; starlike; convex

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Al-Oboudi, F. M., On univalent functions defined by a generalized Salagean operator, Internat. J. Math. Math. Sci. 27 (2004), 1429-1436.

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.

Data publikacji: 2016-07-29 22:06:15
Data złożenia artykułu: 2016-07-29 17:48:23


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