On Certain Classes of Multivalent Analytic Functions

Problem statement: By means of the Hadamard product (or convolution), new class of function of power order was formed. This class was motivated by many authors namely MacGregor, Umezawa, Darus and Ibrahim and many others. The cla ss indeed extended in the form of integral operator due to the work of Bernardi, Libera and Li vingston. Approach: A new class of multivalent analytic functions in the open unit disk U was intr oduced. An application of this class was posed by using the fractional integral operator. The integra l operator of multivalent functions was proposed an defined. The previous well known integral operator was mentioned. Results: Having the integral operator, a class was defined and coefficient bound s established by using standard method. These results reduced to well-known results studied by va rious authors. The operator was then applied for fractional calculus and obtained the coefficient bo unds. Conclusion: Therefore, new operators could be obtained with some earlier results and standard methods. New classes were formed and new results of special cases were obtained.


INTRODUCTION
Let Σ p, α denote the class of functions of the form: which are analytic in the unit disk U. For the Hadamard product or convolution of two power series f defined in (2) and a function g where: p n p n n 2 Note that the authors defined and studied some classes of analytic functions take the form (1) and (2) .

MATERIALS AND METHODS
In this study, we need to introduce a new integral operator such that certain classes can be defined by means of this integral operator. The previous operators will also be mentioned to highlight the importance of simple operator which then can be extended to a complicated ones and yet interesting to study.
Clearly, (4) yields: Thus, by applying the operator J α,p,c successively, we can obtain the multiplier transformation: Cho (1993) defined and studied some subclasses involving the operator k 0,p,c J .
A function f∈Σ p,α is said to be (p+α)-valent starlike of order 0≤µ<p+α if and only if: We denote by S p,α (µ) the class of all such functions. A function f ∈Σ p,α is said to be (p+α)-valent convex of order 0≤µ<p+α if and only if: Let C p,α (µ) denote the class of all those functions.
The main aim of this work is to study coefficient bounds and extreme points of the general subclass of Σ p,α. Furthermore, we obtain special results.

RESULTS
Here we obtain a necessary and sufficient condition and extreme points for functions f ∈Σ p,α .
Theorem 1: Let f∈Σ p,α A sufficient condition for a function of the form (1) to be in S p,α (µ,ν) is that: k n n 2 c p n(1 ) p | a | 1 c p n p (z U) for 0≤µ<p+α, ν≥0.
Proof: Let f be of the form (1). It suffices to show that: Yields: Proof: By letting ν = 0. Next we introduce some well known results which were studied by different authors.

DISCUSSION
The class Σ p,α can further be applied in fractional calculus. For that reason, we need to the following definition.