A NEW SUBCLASS OF BI-UNIVALENT FUNCTIONS DEFINED BY q-DERIVATIVE

Abstract

In this investigation we introduce, by making use of q-derivative operator, a new subclass which are an extension of some well-known subclasses of bi-univalent functions. Also, we give the upper bounds for the coecients ja2j and ja3j for the functions belonging to this new subclass and its subclasses.

Keywords

• q-derivative,bi-univalent function,coecient inequality,subordination.

References

1. Akta¸s ˙I., Baricz ´A., (2017), Bounds for radii of starlikeness of some q−Bessel functions, Results Math., 72, 947-963. Doi: 10.1007/s00025-017-0668-6.
2. Akta¸s ˙I., Orhan H., (2015), Distortion bounds for a new subclass of analytic functions and their partial sums, Bull. Transilv. Univ. Bra¸sov Ser. III: Mathematics, Informatics, Physics., Vol 8(57), No.2 pp. 1–12.
3. Akta¸s ˙I., Orhan H., (2018), On partial sums of normalized q−Bessel functions, Commun. Korean Math. Soc., Vol 33, No.2, pp. 535–547.
4. Ali R. M., Lee S. K., Ravichandran V. and Supramaniam S., (2012), Coefficient estimates for biunivalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25 (3), 344–351.
5. Brannan D. A. and Taha T. S., (1986), On some classes of bi-univalent functions, Studia Universitatis, Babe¸s-Bolyai. Math., 31 (2), 70-77.
6. Duren P. L., (1983), Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, New York.
7. Deniz E., (2013), Certain subclasses of bi-univalent functions satisfying subordinate conditions, Journal of Classical Analysis, vol. 2, number 1, 49–60.
8. Ezeafulukwe U. A. and Darus M., (2015), A note on q−calculus, Fasciculi Mathematici, no. 55, pp. 53–63.

9. Ezeafulukwe U. A. and Darus M., (2015), Certain properties of q−hypergeometric functions, Interna- tional Journal of Mathematics and Mathematical Sciences, vol. 2015, Article ID 489218, 9 pages.
10. Frasin B. A. and Aouf M. K., (2011), New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (9), 1569–1573.
11. Ismail M. E. H, Merkes E. and Styer D., (1990), A Generalization of Starlike Functions, Complex Variables, vol. 14, pp. 77–84.
12. Jackson F. H., (1910), On q− definite integrals, The Quarterly Journal of Pure and Applied Mathe- matics, vol. 41, pp. 193–203.
13. Jackson F. H., (1909), On q−functions and a certain difference operator, Transactions of the Royal Society of Edinburgh, vol. 46, no. 2, pp. 253–281.
14. Lewin M., (1967), On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18, 63–68.
15. Ma W. C. and Minda D, (1992), Unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157–169, Conf. Proc. Lecture Notes Anal. I Int. Press, Cambridge, MA.
16. Orhan H., Toklu E. and Kado˘glu E., (2017), Second Hankel Determinant Problem for k-bi-starlike Functions, Filomat 31:12, 3897–3904.
17. Orhan H., Toklu E. and Kadıo˘glu E., (2018), Second Hankel determinant for certain subclasses of bi-univalent functions involving Chebyshev polynomials, Turk J Math, 42: 1927 1940.
18. Raghavendar K. and Swaminathan A., (2012), Close-to-convexity of basic hypergeometric functions using their Taylor coefficients, J. Math. Appl., 111–125.
19. Seoudy T. M. and Aouf M. F., (2014), Convolution properties for certain classes of analytic functions defined by q−derivative operator, Abstract and Applied Analysis vol. 2014, Article ID 846719, 7 pages.
20. Seoudy T. M. and Aouf M. F., (2016), Coefficient estimates of new classes of q−starlike and q−convex functions of complex order, Journal of Mathematical Inequalities, vol. 10, no. 1, pp. 135–145.
21. Sahoo S. K. and Sharma N. L., (2015), On a generalization of close-to-convex functions, Ann. Polonici Math., 113, 108–205.
22. Srivastava H. M. and Owa S., (1989), Univalent Functions, Fractional calculus, and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto. 68, Number 1, Pages 776–783, DOI: 10.31801/cfsuasmas.475818