Certain subclasses of analytic functions defined by a new general linear operator
Hypergeometric functions are of special interests among the complex analysts especially in looking at the properties and criteria of univalent. Hypergeometric functions have been around since 1900’s and have special applications according to their own needs. Recently, we had an opportunity to study on q-hypergeometric functions and quite interesting to see the behavior of the functions in the complex plane. There are many different versions by addition of parameters and choosing suitable variables in order to impose new set of q-hypergeometric functions. The aim of this paper is to study and introduce a new convolution operator of q-hypergeometric typed. Further, we consider certain subclasses of starlike functions of complex order. We derive some geometric properties like, coefficient bounds, distortion results, extreme points and the Fekete-Szego inequality for these subclasses.
About the Authors
Abdul Rahman Salman Juma, Department of Mathematics, University of Anbar, 55431 Baghdad, 55 Ramadi, Iraq, e-mail: firstname.lastname@example.org
Maslina Darus, Prof. of School of Mathematical Sciences (Faculty of Science and Technology), Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D. Ehsan, Malaysia, e-mail: email@example.com
1. Mohammed A., Darus M. A generalized operator involving the q-hypergeometric function. Matematicki Vesnik, 2013, vol. 65, no. 4, pp. 454-465.
2. Aldweby H., Darus M. Univalence of a new general integral operator associated with the q-hypergeomtric function.Inter. Jour. Math. Sci., ID 769537, 2013, 5 p.
3. Frasin B.A. Family of analytic functions of complex order. Acta Math. Acad. Paedagog. Nyiregyhaziensis, 2006, vol. 22, no. 2, pp. 179-191.
4. Gasper G., Rahman M. Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1990, vol. 35.
5. Srivastava H.M., Choi J. Series Associated with Zeta and Related Functions. Kluwer Academic, Dordrecht, 2001. https://doi.org/10.1007/978-94-015-9672-5
6. Srivastava H.M. Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc., 2000, vol. 129, issue 1, pp. 77-84. https://doi.org/10.1017/S0305004100004412
7. Prajapat J.K., Bulboaca T. Double subordination preserving properties for a new generalized Srivastava-Attiya operator. Chin. Ann. Math. 2012, vol. 33, pp. 569-582. https://doi.org/10.1007/s11401-012-0722-3
8. Noor K.I., Bukhari S.Z.H. Some subclasses of analytic and spiral-like functions of complex order involving the Srivastava-Attiya integral operator. Integral Transforms Spec. Funct., 2010, vol. 21, pp. 907-916.https://doi.org/10.1080/10652469.2010.487305
9. Choi J.H., Saigo M., Srivastava H.M. Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl., 2002, vol. 276, pp. 432-445. https://doi.org/10.1016/S0022-247X(02)00500-0
10. Srivastava H.M., Attiya A.A. An integral operator associated with the Hurwitz-Lerch zeta function and differential subordination. Integral Transforms Spec. Funct., 2007, vol. 18, pp. 207-216. https://doi.org/10.1080/10652460701208577
11. Cho N.E., Srivastava H.M. Argument estimation of certain analytic functions defined by a class of multiplier transformation.Math. Comput. Model., 2003, vol. 37, pp. 39-49. https://doi.org/10.1016/S0895-7177(03)80004-3
12. Salagean S. Subclasses of univalent functions. Lecture Notes in Math., 1983, vol. 1013, pp. 362-372. https://doi.org/10.1007/BFb0066543
13. Bernardi S.D. Convex and starlike univalent functions. Trans. Amer. Math. Soc., 1969, vol. 135, pp. 429-446. https://doi.org/10.1090/S0002-9947-1969-0232920-2
14. Carlson B.C., Shaffer D.B. Starlike and prestarlike hypergeometric functions. SIAM J. Math. Anal., 1984, vol. 15, pp. 737-745. https://doi.org/10.1137/0515057
15. Dziok J., Srivastava H.M. Classes of analytic functions associated with the generalized hypergeometric function.Appl. Math. Comput., 1999, vol. 103, pp. 1-13. https://doi.org/10.1016/S0096-3003(98)10042-5
16. Hohlov Y.E. Operators and operations in the class of univalent functions. Izv. Vys. Ucebn. Zaved. Matematika, 1978, vol. 10, pp. 83-89.
17. Ruscheweyh St. New criteria for univalent functions. Proc. Amer. Math.Soc., 1975, vol. 49, pp. 109-115. https://doi.org/10.1090/S0002-9939-1975-0367176-1
18. Srivastava H.M., Gaboury S. A new class of analytic functions defined by means of a generalization of the Srivastava-Attiya operator. Journal of Inq. and App., 2015, vol. 39.
19. Ma W.C., Minda D. A unified treatment of some special classes of functions. Proceedings of the Conference on Complex Analysis, Tianjin, 1992. Conf. Proc. Lecture Notes in Anal. International Press, Cambridge, 1994, vol. 1, pp. 57-169.
20. Aldweby H., Darus M. A subclass of harmonic univalent functions associated with q-analogue of Dziok-Srivastava operator. ISRN Mathematical Analysis, 2013, vol. 2013, article ID 382312, 6 p.
21. Aldweby H., Darus M. On harmonic meromorphic functions associated with basic hypergeometric functions.The Scientific World Journal, 2013, vol. 2013, article ID 164287, 7 p.
22. Aldweby H., Darus M. Some subordination results on q-analogue of Ruscheweyh differential operator. Abstract and Applied Analysis, 2014, vol. 2014, article ID 958563, 6 p.