Application of Generalized Fractional Integral Operators to Certain Class of Multivalent Prestarlike Functions Defined by the Generalized Operator L λ p ( a , c )

In this paper we introduce certain classes of multivalent prestarlike functions with negative coefficients defined by using the Cho-Kwon Srivastava operator and investigate some distortion theorems in terms of the fractional operator involving H-functions. Classes preserving integral operator and Radius of convexity for this classes and are also included.


Introduction
which are analytic and p-valent in the unit disk U = {z: |z| < 1}.And let T p denote the subclass of A p consisting of analytic and p-valent functions which can be expressed in the form a k+p z k+p , (a k+p ≥ 0). (1.2) A function f (z) ∈ A p is said to be p-valent starlike of order α, if and only if for some α(0 ≤ α < p).We denote the class of all p-valent starlike functions of order α by S p (α).Further a function f (z) from A p is said to be convex of order α if and only if for some α(0 ≤ α < p).We denoted the class of all p-valent starlike functions of order α by C p (α).The classes S p (α) and C p (α) were first introduced by D. A. Patil.

The function
is well-known as the extreme function for the class S p (γ). Sitting then S γ (z) can be written in the form (1.7) We note that C(γ, k) is a decreasing function in γ and that (1.8) Let( f * g)(z) denote the Hadamard product (convolution) of two analytic functions f (z) and g (z), that is, if f (z) is given by (1.1) and g(z) is given by b k+p a k+p z k+p .
The class R p (γ, α) and was studied by M. K Aouf and G. M. Shenen, while the class R p (γ, α) = p (α) is the class of p-valent prestarlike functions of order α and was studied by G. A. Kumar and others.For where and (a) k is the Pochammer symbol defined by In 2004, N. E. Cho introduced the following linear operator L λ p (a, c) analogous to L p (a, c) where φ * p (a, c; z) is the function defined in terms of the Hadamard product (or convolution) by the following condition (1.15) We can easily find from (1.13), (1.14) and (1.15) and for the function f (z) ∈ T p that (1.18) Also by specializing the parameters λ, a, c we obtain from (1.16) and where D n+p−1 is the well-known Ruschewehy derivative of order n + p − 1.
The function f (z) is said to be subordinate to g(z) U written f (z) ≺ g(z) if there exist a function w(z) analytic U such that w(0) = 0, and |w(z Now making use of N. E. Cho operator L λ p (a, c) defined by (1.16) we introduce the following subclass S λ p (a, c, A, B, γ, α) of p-valent γ prestarlike function of order α(0 ≤ α < p; 0 ≤ γ < p).
We note that: generalizes and extends other classes studied and introduced by several researches as M. K. Aouf, G. A. Shenan, G. A. Kumar, T. Sheil-Small, and S. Owa.

Fractional Integral Operators
We shall be concerned with fractional integral operators involving Fox's H-function which has been recently introduced by S. L. Kalla and V. S. Kiryakova.
Definition 2.1 Let s ∈ N 0 (the set of non-negative integers), β j ∈ R + (the set of positive real numbers) and δ j , γ j∈C (the set of complex numbers) for j = 1, 2, ..., s, while ∞ k=1 Re(δ j ) > 0, then the generalized fractional integral operator for the function f (z) is given by where f (z) is an analytic function in a simply connected region of the z-plane containing the origin and where a j = γ j + δ j + 1 − t/β j and b j = γ j + 1 − t/β j ( j = 1, 2, ..., s).
The fractional integral operator I (γ s ;(δ s ) (β s );s f (z) contains as special case, many other fractional integral operators (M.Saigo), here we need some of them.
(i) For t > 0 and f (z) being analytic function in a simply connected region of the z-plane containing the origin, Where Re(β) > 0 and the multiplicity of (z − t) β−1 is remove by requiring log|z − t| to be real when (z − t) > 0, D −β z f (z) is called the Riemann-Liouville fractional integral operator of order β (S.G. Samko).(ii) With Re(δ) > 0, β, η ∈ C and f (z) being analytic function in a simply connected region of the z− plane containing the origion, (2.4) where I δ,β,η 0,z f (z)is the known Saigo fractional integral operator δ, (M.Saigo) and with the order and the multiplicity of (Z − 1) δ−1 is removed as in M. Saigo.
Proof.Let the function f (z) ∈ T P defined by (1.2), then from (1.7) and (1.16) we have assume that the inequality (3.1) holds true and let |z| = 1, then from (1.21) we have Hence by the principle of maximum modulus f (z) ∈ T λ p (a, c, A, B, γ, α).Conversely, assume that f (z) defined by (3.1) is in the class T λ p (a, c, A, B, γ, α), then from (3.3) we have since |Re(z)| ≤ |z| for all z, we have choose the values z on the real axis so that Upon clearing the denominator of (3.4) and letting z → 1 through real values we get which implies the inequality (3.1).Sharpness of the result follows by setting 4. Distortion Theorem for the Class T λ p (a, c, A, B, γ, α) Theorem 4.1 Let β j ∈ N, δ j ∈ R, ( j = 1, 2, 3, ..., s) be such that Re(γ j ) > −Re(pβ j − 1), and s j=1 For z ∈ U, the equalities in (4.2) and (4.3) are attained by the function Proof.Making use of (1.2) and Lemma 2.1, we obtain From (2.6) we have where Under the assumption of the theorem we see that ψ(k) is non-increasing on k, i.e.
Also, we have which implies the assertion (4.3) of Theorem (4.1).
Corollary 4.1 Let the function f (z) defined by (1.2) be in the class T λ p (a, c, A, B, γ, α), then For z ∈ U, the equalities in (4.9) and (4.10) are attained by the function given by (4.4).
Corollary 4.2 Let the function f (z) defined by (1.2) be in the class T λ p (a, c, A, B, γ, α), then under the assumption (4.12) For (z ∈ U 0 ), where The equality in (4.11) and (4.12) are attained by the function given by (4.4).
Corollary 4.3 Let the function f (z) defined by (1.2) be in the class T λ p (a, c, A, B, γ, α), then where σ is a positive integer and z ∈ U 0 , where The equality in (4.13) and (4.14) are attained by the function given by (4.4).
) and using (2.5) we obtain the result.

Remark 4. 3
Several other particular studied cases studied by different authors can be obtained from Theorem (4.1) by specializing the parameters s, β s , δ s , γ, α, γ, λ, A, B, a and c see for example H. M. Srivastava and P. K. Banerji.5.Integral Transform of the Class T λp (a, c, A, B, γ, α) The generalized Komatu integral operator L δ c,p : T P → T P is defined for δ > 0 and c > −p as (see, e.g. S. M. Khairnar and T. O. Salim)H(z) = L δ c,p f (z) = (c + p) δ Γ(δ)z c z 0 t c−t log z t δ−1 f (t)dt.