Subclasses of Starlike Functions Associated with Fractional -Calculus Operators

Making use of fractional -calculus operators, we introduce a new subclass of starlike functions and determine the coefficient estimate, extreme points, closure theorem, and distortion bounds for functions in . Furthermore we discuss neighborhood results, subordination theorem, partial sums, and integral means inequalities for functions in . 1. Introduction and Preliminaries Denote by the class of functions of the form which are analytic and univalent in the open disc and normalized by . Due to Silverman [1], denote by a subclass of consisting of functions of the form The fractional calculus operator has gained importance and popularly due to vast potential demonstrated applications in various fields of science, engineering and also in the geometric function theory. The fractional -calculus operator is the extension of the ordinary fractional calculus in the -theory. Recently Purohit and Raina [2] investigated applications of fractional -calculus operator to define new classes of functions which are analytic in the open unit disc. We recall the definitions of fractional -calculus operators of complex valued function . The -shifted factorial is defined for as a product of factors by and in terms of basic analogue of the gamma function Due to Gasper and Rahman [3], the recurrence relation for -gamma function is given by and the binomial expansion is given by Further the -derivative and -integral of functions defined on the subset of are, respectively, given by It is interest to note that the familiar Pochhammer symbol. Due to Kim and Srivastava [4], we recall the following definitions of fractional -integral and fractional -derivative operators, which are very much useful for our study. Definition 1. Let the function be analytic in a simply connected region of the -plane containing the origin. The fractional -integral of of order is defined by where can be expressed as the -binomial given by (6) and the series is a single valued when and , therefore the function in (8) is single valued when , and . Definition 2. The fractional -derivative operator of order is defined for a function by where the function is constrained, and the multiplicity of the function is removed as in Definition 1. Definition 3. Under the hypothesis of Definition 2, the fractional derivative of order is defined by With the aid of the above definitions, and their known extensions involving -differintegral operator we define the linear operator where where , and . Here in (10) represents, respectively, a fractional -integral of of order when and fractional -derivative of of order when .