Generalization of Certain Subclasses of Multivalent Functions with Negative Coefficients Defined by Cho-Kwon-Srivastava Operator

Making use of the Cho-Kwon-Srivastava operator, we introduce and study a certain SCn (j, p, λ, α, δ) of p-valently analytic functions with negative coefficients. In this paper, we obtain coefficient estimates, distortion theorem, radii of close-to-convexity, starlikeness, convexity and modified Hadamard products of functions belonging to the class SCn (j, p, λ, α, δ). Finally, several applications investigate an integral operator, and certain fractional calculus operators also considered.


Introduction
 is said to be p-valently starlike of order  if it satisfies the inequality : We denote by the class of all p-valently starlike functions of order  .Also a function   f z  is said to be p-valently convex of order  , T j p  if it satisfies the inequality:  are studied by Owa [3].
In [4] Wang et al. defined Cho-kwon-Srivastava operator which is the well-known Cho-kwon-Srivastava operator (see [5]) where   the class of all p-valently convex functions of order  .We note that (see for example Duren [1] and Goodman [2]) , We note that: 1) when 0, which is the class of starlike of order .


2) when 0 , we have the class which is the class of starlike functions of order  studied by Owa [3] and Yamakawa [6] 3) when 1, which is the class of convex functions of order  studied by Owa [3] and Yamakawa [6].
In our present paper, we shall make use of the familiar , c p J defined by (c.f.[7,8], see also [9]) as well as the fractional calculus operator z D  for which it is well known that (see, for details, [10,11]; see also Section 5 below) in terms of Gamma functions.

Coefficient Estimates
Proof.Assume that the inequality (2.1) holds true.Then we have This shows that the values of the function , lie in a circle which is centered at and whose radius is for some some , p, Thus we have the inequality (2.1).

Corollary 1. Let the function  
f z defined by (1.1) be in the class

 
, , , , The result is sharp for the function   The result is sharp for the function   f z given by Proof.In view of Theorem 1, we have Now, by differentiating both sides of (1.1) m times, we obtain

 
, , , , Each of these results is sharp for the function   f z given by (2.7).
Proof.It is sufficient to show that and for a function  where 1 2 and 3 are defined by (4.1) -( 4.3) respectively.The details involved are fairly straightforward and may omi-, r r r tited.

Modified Hadamard Products
For the functions we denote by    Theorem 4. Let the functions defined by (5.1) be in the class , , , , 3 The result is sharp for the functions given by .
Proof.Employing the technique used earlier by Schild and Silverman [12], we need to find the largest γ such that Therefore, by the Cauchy-Schwarz inequality, we obtain Thus we only need to show that , , , or, equivalently, that , , , Hence, in light of the inequality (5.7), it is sufficient to prove that It follows from (5.10) that Now, defining the function (5.12) We see that is an increasing function of k.Therefore, we conclude that which evidently completes the proof of Theorem 4.
we obtain Corollary 2. Let the functions defined by (5.1) be in the class The result is sharp.Corollary 3. Let the functions defined by (5.1) be in the class .

  
The result is sharp.Using arguments similar to those in the proof of , , , , The result is the best possible for the functions , , , , n j p defined by (5.1) be in the class SC    .Then the function (5.21) The result is sharp for the functions given by (5.4).Proof.Noting that , Therefore, we have find largest  such that Now, defining the function by we observe that is an increasing function of k.We thus conclude that which completes the proof of Theorem 6.
For our present investigation, we recall the following definitions.
Definition 1.The fractional integral of order  is defined, for a function   f z , by where the function   f z is analytic in a simply-connected domain of the complex z-plane containing the origin and the multiplicity of   where the function   In this section, we shall investigate the growth and distortion properties of functions in the class  , , , z D  In order to derive our results, we need the following Lemma given by Chen et al. [14].
(6.7)Each of the assertion (6.6) and (6.7) is sharp.Proof.In view of Theorem 1, we have which readily yields Thus, by using (6.9) and (6.11), we deduce that  which yield the inequalities (6.6) and (6.7) of Theorem 7.
The equalities in (6.6) and (6.7) are attained for the func-tion   f z given by or, equivalently, by  Using arguments similar to those in the proof of Theorem 7, we obtain the following result.

 
and Each of the assertions (6.14) and (6.15) is sharp.
and p-valent in the open unit disc    .Then we have

4 )
Theorem 2, follows from (3.3) and(3.4).Finally, it is easy to see that the bounds in (3.1) are at-tained for the function   f z given by (3.2).

4 .Theorem 3 .
Radii of Close-to-Convexity, Starlikeness and Convexity Let the function   f z defined by (1.1) be in the class

Theorem 4 , 5 .
we obtain the following result.Theorem Let the function

Definition 3 .
Under the hypotheses of Definition 2, the fractional derivative of order n   is defined, for a function   f z , by

Theorem 7 .
Let the function   f z defined by (1.1) be in the class

6 . 13 ) 8 .
Theorem Let the function   f z defined by (1   Then Thus we complete the proof of Theorem 7.