The study of the operators possessing convolution form and their properties is considered advantageous in geometric function theory. In 1975 Ruscheweyh defined operator for analytic functions using the technique of convolution. In 2005, Goyal and Goyal generalized the Ruscheweyh operator to fractional order (which we call here Ruscheweyh-Goyal differential operator) using Srivastava-Saigo fractional differential operator involving hypergeometric function. Inspired by these earlier efforts, we discuss the properties of the Ruscheweyh-Goyal derivative of arbitrary order. We define a class of pre-starlike type functions involving the Ruscheweyh-Goyal fractional derivative and obtain the inclusion relation. Further, we prove that Ruscheweyh-Goyal derivative operator preserve the convexity and starlikeness for an analytic function. The majorization results for fractional Ruscheweyh-Goyal derivative has been discussed using a newly defined subclass.
Agarwal, Ritu and Paliwal, G. S.
Ruscheweyh-Goyal Derivative of Fractional Order, its Properties Pertaining to Pre-starlike Type Functions and Applications,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 15,
3, Article 8.
Available at: https://digitalcommons.pvamu.edu/aam/vol15/iss3/8