Ruscheweyh Derivative and a New Generalized Operator Involving Convolution

**MLA Style: **Ezekiel Abiodun Oyekan, S.R. Swamy, Timothy Oloyede Opoola "Ruscheweyh Derivative and a New Generalized Operator Involving Convolution" International Journal of Mathematics Trends and Technology 67.1 (2021):88-100.

**APA Style: **Ezekiel Abiodun Oyekan, S.R. Swamy, Timothy Oloyede Opoola(2021). Ruscheweyh Derivative and a New Generalized Operator Involving Convolution International Journal of Mathematics Trends and Technology, 88-100.

**Abstract**

In this present investigation, we introduce a generlized differential operator E^{m}_{α,β}(μ,Φ,t) via convolution approach. Using this operator, we further introduced a new generalized differential operator RE^{m}_{α,β,δ}(σ,φ,t)f(Z) obtained as a linear combination of Ruscheweyh derivative and the operator E^{m}_{α,β}^{(μ,Φ,t). With the aid of the new generalized differential operator, a new subclass Mm,λ,μ,σ,φ,tα,β,δ (ρ) of analytic functions in the open unit disk is introduced and investigated. Characterization and other properties of this class are studied. In particular, Coefficient estimates, distortion theorems of functions with negative coefficients belonging to this class are also determined. Some relevant remarks and useful connections of the main results are also pointed out. }_{ } ^{ }_{ }

**Reference**

[1] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49 (1975) 109-115.

[2] S. R. Swamy, Inclusion properties of certain subclasses of analytic functions, Int. Math. Forum, 7(36) (2012) 1751-1760.

[3] F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci., 27 (2004) 1429-1436.

[4] A. Catas, On certain class of p-valent functions defined by new multiplier transformations, Proceedings book of the an international symposium on geometric function theory and applications, August, 20-24, 2007, TC Istanbul Kultur Univ., Turkey, 241-250.

[5] N. E. Cho and H. M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modeling, 37(1-2) (2003) 39-49.

[6] N. E. Cho and T. H. Kim, Multiplier transformations and strongly Close-to Convex functions, Bull. Korean Math. Soc., 40(3) (2003) 399-410.

[7] G. St.Salagean, Subclasses of univalent functions, Proc. Fifth Rou. Fin. Semin. Buch. Complex Anal., Lect. notes in Math., Springer -Verlag, Berlin, 1013(1983), 362-372.

[8] S. S. Bhoosnurmath and S. R. Swamy, On certain classes of analytic functions, Soochow J. Math., 20 (1) (1994) 1-9.

[9] S. R. Swamy, A note on a subclass of analytic functions defined by the Ruscheweyh derivative and a new generalized multiplier transformation, J. Math. Computational Sci., 2(4) (2012) 784-792.

[10] S. R. Swamy, New classes containing Ruscheweyh derivative and a new generalized multiplier differential operator, American International Journal of Research in Science, Technology, Engineering & Mathematics, 11(1) (2015) 65-71.

[11] S. R. Swamy, On analytic functions defined by Ruscheweyh derivative and a new generalized multiplier differential operator, Inter. J. Math. Arch., 6 (7) (2015) 168-177.

[12] S. R. Swamy, Subordination and superordination results for certain subclasses of analytic functions defined by the Ruscheweyh derivative and a new generalized multiplier transformation, J. Global Res. Math. Arch., 1 (6) (2013) 27-37.

[13] S. R. Swamy, Ruscheweyh Derivative and a New Generalized Multiplier Differential Operator, Annals of Pure and Applied Mathematics, 10(2), (2015), 229-238.

[14] A. Alb Lupas and L. Andrei, New classes containing generalized Salagean operator and Ruscheweyh derivative, Acta Univ. Apulensis, 38 (2014) 319-328.

[15] O. T. Opoola, On a subclass of Univalent Functions defined by a Generalized Differential operator, Int. J. Math. Anal. 18(11) (2017) 869-876.

[16] A. W. Goodman, Univalent functions. Vol. I, Mariner, Tampa, FL, 1983.

[17] P. L. Duren, Univalent functions, Springer, New York, 1983.

[18] S. Ruscheweyh, Convolutions in geometric function theory, Presses Univ. Montréal, Montreal, Que., 1982.

[19] E. A. Oyekan and I. T. Awolere, A new subclass of univalent functions connected with convolution defined via employing a linear combination of two generalized differential operators involving sigmoid function, Maltepe J. Math., Vol. II Issue 2 (2020), 82-96

[20] M. Darus and R. Ibrahim, New classes containing generalization of differential operator, Applied Math. Sci., 3 (51) (2009) 2507-2515.

[21] H. Silverman, Univalent functions with negative coefficients, Proc, Amer. Math. Soc., 51(1) (1975) 109-116.

**Keywords : **analytic function, differential operator, distortion theorem, multiplier transformation.