Certain class of analytic functions involving Salagean type q-difference operator

Year 2018, Volume: 6 Issue: 2, 264 - 271, 15.10.2018

Sibel Yalçın , Kaliappan Vijaya Gangadharan Murugusundaramoorthy

Abstract

In
this paper, we define a new subclass of analytic functions with negative coefficients
involving Salagean type q-difference operator and discuss certain characteristic
properties and inclusion relations involving N_δ(e) of this generalized function
class. Further, we determine partial sums results for the function class. The
usefulness of the main result not only provide the unification of the results
discussed in

the literature but also generate certain new results.

Keywords

Analytic, Univalent, q-difference operator, Partial sums

References

• [1] S¸ . Altınkaya, and S. Yalc¸ın, Faber polynomial coefficient estimates for a class of bi-univalent functions based on the symmetric q-derivative operator, Journal of Fractional Calculus and Applications, Vol:8, No:2 (2017), 79-87.
• [2] S¸ . Altınkaya, and S. Yalc¸ın, On the Fekete-Szeg¨o problem for analytic functions defined by using symmetric q-derivative operator, Konuralp Journal of Mathematics, Vol:5, No:1 (2017), 176-186.
• [3] S. Araci, U. Duran, M. Acikgoz and H. M. Srivastava, A certain (p, q)-derivative operator and associated divided differences, J. Inequal. Appl., (2016) 2016:301.
• [4] Aral, A., Gupta, V. and Agarwal, R. P., Applications of q-calculus in operator theory, Springer, New York, 2013.
• [5] B.A. Frasin, Partial sums of certain analytic and univalent functions, Acta Math. Acad. Paed. Nyir. Vol:21 (2005), 135-145.
• [6] B.A. Frasin and G.Murugusundaramoorthy Partial sums of certain analytic functions, Mathematica,Tome 53, Vol:76, No:2,(2011), 131-142.
• [7] A.W.Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc., Vol:8 (1957), 598-601.
• [8] F. H. Jackson, On q-functions and a certain difference operator, Transactions of the Royal Society of Edinburgh, Vol:46 (1908), 253-281.
• [9] S. Kanas and D. R˘aducanu, Some subclass of analytic functions related to conic domains, Math. Slovaca, Vol:64, No:5 (2014), 1183-1196.
• [10] Z. Karahuseyin, S¸ . Altınkaya and S. Yalc¸ın, On H3(1) Hankel determinant for univalent functions defined by using q􀀀derivative operator, Transylv. J. Math. Mech., Vol:9, No:1 (2017), 25-33.
• [11] M.Govindaraj, and S. Sivasubramanian, On a class of analytic function related to conic domains involving q􀀀calculus, Analysis Math., Vol:43, No:5 (2017), 475-487.
• [12] G.Murugusundaramoorthy and H. M. Srivastava, Neighborhoods of certain classes of analytic functions of complex order, Journal of Ineql. In Pure and Appl. Maths., Vol:5, No:2, Art.24 (2004) 1-8.
• [13] S. D. Purohit, and R. K. Raina, Fractional q-calculus and certain subclasses of univalent analytic functions, Mathematica Vol:55(78), No:1 (2013), 62-74.
• [14] T. Rosy, K.G. Subramanian and G. Murugusundaramoorthy, Neighborhoods and partial sums of starlike based on Ruscheweyeh derivatives, J. Ineq. Pure Appl. Math., Vol:4, No:4, Art. 64 (2003), 1-8.
• [15] S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc., Vol:81 (1981), 521-527.
• [16] Salagean, G. S., Subclasses of univalent functions, Lecture Notes in Math., Springer-Verlag, 1013 (1983), 362-372.
• [17] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., Vol:51 (1975), 109-116.
• [18] H. Silverman, Partial sums of starlike and convex functions, J.Math Anal.&.Appl., Vol:209 (1997), 221–227.
• [19] T. Sheill-Small, A note on partial sums of convex schlicht functions, Bull. London Math. Soc., Vol:2 (1970), 165-168.
• [20] E.M. Silvia., Partial sums of convex functions of order a, Houston.J.Math., Vol:11, No:3 (1985), 397-404.
• [21] H. M. Srivastava, Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inf. Sci. Vol:5, No:3 (2011), 390-444.
• [22] K.Vijaya, Certain class of analytic functions based on q-difference operator (communicated)