Some properties for $\alpha$-starlike functions with respect to $k$-symmetric points of complex order

H. E. Darwish; A. Y. Lashin; S. M. Sowileh

doi:10.17951/a.2017.71.1.1

Some properties for $\alpha$ -starlike functions with respect to $k$ -symmetric points of complex order

H. E. Darwish, A. Y. Lashin, S. M. Sowileh

Abstract

In the present work, we introduce the subclass

$\mathcal{T}_{\gamma ,\alpha }^{k}(\varphi )$ , of starlike functions with respect to

$k$ -symmetric points of complex order

$\gamma$ (

$\gamma \neq 0$ ) in the open unit disc

$\vartriangle$ . Some interesting subordination criteria, inclusion relations and the integral representation for functions belonging to this class are provided. The results obtained generalize some known results, and some other new results are obtained.

Keywords

Analytic functions; starlike functions of complex order; convex functions of complex order;

$\alpha$ -starlike;

$k$ -symmetric points; subordination

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References

Abubaker, A. A. A., Darus, M., On starlike and convex functions with respect to k-symmetric points, Internat. J. Math. Math. Sci., 2011 (2011), Art. ID 834064, 9 p.

Al-Oboudi, F. M., On classes of functions related to starlike functions with respect to symmetric conjugate points defined by a fractional differential operator, Complex Anal. Oper. Theory 5 (2011), 647-658.

Altintas, O., Irmak, H., Owa, S., Srivastava, H. M., Coefficient bounds for some families of starlike and convex functions of complex order, Appl. Math. Letters 20 (2007), 1218-1222.

Chand, R., Singh, P., On certain schlicht mappings, Indian J. Pure Appl. Math. 10 (9) (1979), 1167-1174.

Das, R. N., Singh, P., On subclasses of schlicht mapping, Indian J. Pure Appl. Math. 8 (1977), 864-872.

Eenigenburg, P., Miller, S. S., Mocanu, P. T., Reade, M. O., On a Briot-Bouquet differential subordination, Rev. Roumaine Math. Pures Appl. 29 (1984), 567-573.

Ma, W. C., Minda, D. A., A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Yang, and S. Zhang (Eds.), Int. Press, 1994, 157-169.

Nasr, M. A., Aouf, M. K., On convex functions of complex order, Mansoura Sci. Bull. Egypt. 9 (1982), 565-582.

Nasr, M. A., Aouf, M. K., Starlike functions of complex order, J. Natur. Sci. Math. 25 (1985), 1-12.

Padmanabhan, K. S., Parvatham, R., Some applications of differential subordination, Bull. Aust. Math. Soc. 32 (3) (1985), 321-330.

Parvatham, R., Radha, S., On (alpha)-starlike and (alpha)-close-to-convex functions with respect to n-symmetric points, Indian J. Pure Appl. Math. 16 (9) (1986), 1114-1122.

Pascu, N. N., Alpha-close-to-convex functions, in: Romanian–Finnish Seminar on Complex Analysis (Bucharest, 1976), Springer, Berlin, 1979, 331-335.

Pascu, N. N., Podaru, V., On the radius of alpha-starlikeness for starlike functions of order beta, in: Complex analysis – fifth Romanian–Finnish seminar, Part 1 (Bucharest, 1981), Springer, Berlin, 1983, 336–349.

Ramachandran, C., Kavitha, D., Soupramanien, T., Certain bound for q-starlike and q-convex function with respect to symmetric points, Internat. J. Math. Math. Sci. 2015 (2015), Art. ID 205682, 7 pp.

Ravichandran, V., Polatoglury, Y., Bolcaly, M., Seny, A., Certain subclasses of starlike and convex functions of complex order, Hacettepe J. Math. Stat. 34 (2005), 9-15.

Robertson, M. S., On the theory of univalent functions, Ann. Math. 37 (1936), 374-408.

Sakaguchi, K., On certain univalent mapping, J. Math. Soc. Japan 11 (1959), 72-75.

Wang, Z.-G., Gao, C.-Y., Yuan, S.-M., On certain subclasses of close-to-convex and quasi-convex functions with respect to k-symmetric points, J. Math. Anal. Appl. 322 (1) (2006), 97-106.

Wang, Z.-G., Gao, C.-Y., Orhan, H., Akbulut, S., Some subclasses of close-to-convex and quasi-convex functions with respect to k-symmetric points, General Math., 15 (4) (2007), 107-119.

Wiatrowski, P., The coefficient of a certain family of holomorphic functions, Zeszyty Nauk. Uniw. Łódz. Nauki Mat. Przyrod. Ser. II No. 39 Mat. (1971), 75-85.

Yuan, S.-M., Liu, Z.-M., Some properties of -convex and -quasiconvex functions with respect to n-symmetric points, Appl. Math. Comput. 188 (2) (2007), 1142-1150.

DOI: http://dx.doi.org/10.17951/a.2017.71.1.1
Date of publication: 2017-06-30 17:33:53
Date of submission: 2017-06-29 18:27:25

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Some properties for α\alpha-starlike functions with respect to kk-symmetric points of complex order