Certain properties for spiral-like functions associated with Ruscheweyh-type q-difference operator

Abstract

In this paper, making use of the Ruscheweyh- type $q$-difference operator $\mathcal{D}_q(\mathcal{R}_q^\alpha f(z))$ we introduce a new subclass of spiral-like functions and discuss some subordination results and Fekete-Szeg\"{o} problem for this generalized function class. Further, some known and new results which follow as special cases of our results are also mentioned.

Keywords

• Univalent functions,starlike functions,convex functions,spirallike functions,subordinating factor sequence,Fekete-Szeg\"{o} problem

References

1. [1] S. Araci, U. Duran, M. Acikgoz, H. M. Srivastava, A certain (p, q)-derivative operator and associated divided differences, J. Inequal. Appl., (2016) 2016:301.
2. [2] A. Aral, V. Gupta, R. P. Agarwal, Applications of q-calculus in operator theory, Springer, New York, 2013.
3. [3] F. H. Jackson, On q-functions and a certain difference operator, Trans. R. Soc. Edinburgh, 46 (1908), 253-281.
4. [4] S. Kanas, D. R˘aducanu, Some subclass of analytic functions related to conic domains, Math. Slovaca, 64(5)(2014), 1183-1196.
5. [5] F. R. Keogh, E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8-12.
6. [6] R. J. Libera, Univalent a􀀀spiral functions, Canad. J. Math., 19 (1967), 449-456.
7. [7] A. Mohammed, M. Darus, A generalized operator involving the q-hypergeometric function, Mat. Vesnik, 65(4) (2013), 454-464.
8. [8] S. Mahmood, J. Sok´ol, New subclasses of analytic functions in conical domain associated with Ruscheweyh q-differential operator, Results Math., 71 (2017), 1345-1357.

9. [9] Z. Nehari, Conformal mapping, McGraw-Hill, New-York, 1952.
10. [10] H. Orhan, D. R˘aducanu, M. Caglar, M. Rayram, Coefficient estimates and other properties for a class of spirallike functions associated with a differential operator, Abstr. Anal. Appl., vol.2013, Art. ID415319, 7pp.
11. [11] S. D. Purohit, R. K. Raina, Fractional q-calculus and certain subclasses of univalent analytic functions, Mathematica, 55(78), no.1 (2013), 62-74.
12. [12] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49 (1975), 109-115.
13. [13] H. Silverman, Sufficient conditions for spiral-likeness, Internat. J. Math. Sci., 12(4) (1989), 641-644.
14. [14] S. Singh, A subordination theorem for spiral-like functions, Internat. J. Math. Math. Sci., 24(7) (2000), 433-435.
15. [15] L. ˇSpaˇcek, Contribution ` a la theorie des fonctions univalents, Cas. Mat. Fys., 62(2) (1932), 12-19.
16. [16] H. M. Srivastava, A. K. Mishra, M. K. Das, The Fekete-Szeg¨o problem for a subclass of close-to-convex functions, Complex Var. Theory Appl., 44 (2001), 145-163.
17. [17] H. S. Wilf, Subordinating factor sequence for convex maps of the unit circle, Proc. Amer. Math. Soc., 12 (1961), 689-693.