some applications of generalized fractional calculus ...

TAIWANESE JOURNAL OF MATHEMATICS Vol. 8, No. 3, pp. 443-452, September 2004 This paper is available online at http://www.math.nthu.edu.tw/tjm/

SOME APPLICATIONS OF GENERALIZED FRACTIONAL CALCULUS OPERATORS TO A NOVEL CLASS OF ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS Hu¨ seyin Irmak and R. K. Raina Abstract. In the present paper, by making use of certain operators of generalized fractional calculus, we introduce a novel class Tλµ,ϕ,η (n; α) of functions which are analytic and univalent in the open unit disk U. A necessary and sufficient condition for a function to be in the class Tλµ,ϕ,η (n; α) is obtained. In addition, this paper includes distortion theorems involving generalized fractional integrals (and generalized fractional derivatives), radii of close-to-convexity, starlikeness, and convexity. Relevance with some new (or known) special cases are also pointed out.

1. INTRODUCTION AND DEFINITIONS Let T (n) denote the class of functions f (z) of the form: ∞ ak z k , (ak ≥ 0; n ∈ N = {1, 2, 3, · · · }), (1.1) f (z) = z − k=n+1

which are analytic and univalent in the unit open disk U = {z : z ∈ C and |z| < 1}. Tλµ,ϕ,η (n; α)

the subclass of functions f (z) ∈ T (n) which also satisfy We denote by the inequality: 1+µ,1+ϕ,1+η 2 2+µ,2+ϕ,2+η zJ {f (z)} + λz J {f (z)} 0,z 0,z − (1 − ϕ) (1.2) < α, µ,ϕ,η 1+µ,1+ϕ,1+η (1 − λ)J0,z {f (z)} + λzJ0,z {f (z)} (z ∈ U; n ∈ N ; 0 < α ≤ 1; 0 ≤ µ < 1; ϕ, η ∈ R; ϕ < 1; η > max{µ, ϕ} − 2). Received June 15, 2002; revised November 5, 2002. Communicated by H. M. Srivastava. 2000 Mathematics Subject Classification: 30C45, 26A33. Key words and phrases: Open unit disk, maximum modulus theorem, analytic functions, (generalized) fractional integrals, (generalized) fractional derivatives, distortion theorems, starlike and convex functions.

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Hu¨ seyin Irmak and R. K. Raina

µ,ϕ,η Here and throughout this paper, J0,z denotes an operator of fractional calculus, which is defined as follows (cf., [2] and [4]):

Definition 1. The fractional integral of order µ of a function f (z) is defined by z 1 f (ξ) dξ, (µ > 0), (1.3) Dz−µ {f (z)} = Γ(µ) 0 (z − ξ)1−µ where f (z) is analytic in a simply- connected region of the z− plane containing the origin, and the multiplicity of (z − ξ)µ−1 is removed by requiring log(z − ξ) to be real when z − ξ > 0. Definition 2. The fractional derivative of order µ of a function f (z) is defined by z d 1 f (ξ) (1.4) Dzµ {f (z)} = dξ, (0 ≤ µ < 1), Γ(1 − µ) dz 0 (z − ξ)µ where f (z) it is choosen as in (1.3), it and the multiplicity of (z − ξ) −µ is removed, as in Definition 1. Definition 3. Let µ > 0 and η, β ∈ R. Then, in terms of familiar (Gauss’s) µ,β,η of a hypergeometric function 2 F1 , the generalized fractional integral operator I0,z function f (z) is defined by z −µ−β z ξ µ,β,η µ−1 dξ, (z − ξ) f (ξ) ·2 F1 µ + β, −η; 1 − (1.5) I0,z {f (z)} = Γ(µ) 0 z where the function f (z) is analytic in a simply-connected region of the z− plane containing the origin, with the order (1.6)

f (z) = O(|z|), (z → 0),

for (1.7)

> max{0, β − η} − 1,

and the multiplicty of (z − ξ)µ−1 is removed by requiring log(z − ξ) to be real when z − ξ > 0. Definition 4. Let 0 ≤ µ < 1 and η, β ∈ R. Then, the generalized fractional µ,β,η of a function f (z) is defined by derivative operator J0,z

(1.8)

1 µ,β,η {f (z)} = J0,z Γ(1 − µ) z ξ d µ−β −µ z dξ , (z − ξ) f (ξ) ·2 F1 β − µ, 1 − η; 1 − µ; 1 − · dz z 0

Some Applications of Generalized Fractional Calculus

445

where the function f (z) is analytic in a simply-connected region of the z− plane containing the origin, with the order as given by (1.6), and the multiplicity of (z − ξ)−µ is removed by requiring log(z − ξ) to be real when z − ξ > 0. By comparing (1.3) with (1.5), we obtain the following relationship: (1.9)

µ,−µ,η {f (z)} = Dz−µ {f (z)} (µ > 0). I0,z

Similarly, by comparing (1.4) with (1.8), we find (1.10)

µ,µ,η J0,z {f (z)} = Dzµ {f (z)} (0 ≤ µ < 1).

From the general class Tλµ,ϕ,η (n; α) defined by (1.2), we obtain the following important subclasses: (1.11)

V µ,ϕ,η (n; α) = T0µ,ϕ,η (n; α) (n ∈ N ; 0 ≤ µ < 1; 0 < α ≤ 1),

(1.12)

W µ,ϕ,η (n; α) = T1µ,ϕ,η (n; α) (n ∈ N ; 0 ≤ µ < 1; 0 < α ≤ 1),

(1.13)

Ωµ (n; α) = V µ,µ,η (n; α) (n ∈ N ; 0 ≤ µ < 1; 0 < α ≤ 1),

(1.14)

µ (n; α) = W µ,µ,η (n; α) (n ∈ N ; 0 ≤ µ < 1; 0 < α ≤ 1),

(1.15)

Sn (α) = Ω0 (n; α) (n ∈ N ; 0 < α ≤ 1),

(1.16)

Cn (α) = 0 (n; α) (n ∈ N ; 0 < α ≤ 1),

(1.17)

S ∗ (α) = S1 (α) (0 < α ≤ 1),

(1.18)

C ∗ (α) = C1 (α) (0 < α ≤ 1).

The classes S n (α) and S ∗ (α) consist of starlike functions, of order 1 − α (0 ≤ α < 1) and the classes C n (α) and C ∗ (α) consist of convex functions, of order 1 − α (0 ≤ α < 1). These classes are of much importance in the Geometric Function Theory (cf., [9]). Some other interesting papers involving fractional calculus operators are the ones by Altintas¸ et al. ([1], [2]), Chen et al. ([3], [4]), Irmak [6], and Raina and Srivastava [7]. 2. SOME PROPERTIES OF THE CLASS Tλµ,ϕ,η (n; α) We begin by establishing a necessary and sufficient condition for a function f (z) ∈ T (n) to be in the class T λµ,ϕ,η (n; α). This result is contained in the following:

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Theorem 1. Let a function f (z) ∈ T (n). Then, the function f (z) belongs to the class T λµ,ϕ,η (n; α) if and only if ∞ (k + α − 1)Γ(k + 1)Γ(k − ϕ + η + 1)[1 + λ(k − ϕ − 1)] ak Γ(k − ϕ + 1)Γ(k − µ + η + 1)

(2.1)

k=n+1

α(1 − λϕ)Γ(2 − ϕ + η) , Γ(2 − ϕ)Γ(2 − µ + η)

≤

(n ∈ N ; 0 < α ≤ 1; 0 ≤ µ < 1; ϕ, η ∈ R; ϕ < 1; η > max{µ, ϕ} − 2). The result is sharp for the function f (z) given by f (z) = z − (2.2) ·

α(1 − λϕ)Γ(2 − ϕ + η)Γ(n − ϕ + 2) (n + α)[1 + λ(n − ϕ)]Γ(2 − ϕ)Γ(2 − µ + η)

Γ(n − µ + η + 2) z n+1 , Γ(n + 2)Γ(n − ϕ + η + 2)

(n ∈ N ).

Proof. Suppose that the function f (z) is defined by (1.1), and the inequality (2.1) holds true. Then, invoking [9, p. 15, Eq. (2.2)], we have 1+µ,1+ϕ,1+η 2+µ,2+ϕ,2+η {f (z)} + λz 2 J0,z {f (z)} |zJ0,z µ,ϕ,η 1+µ,1+ϕ,1+η {f (z)} + λzJ0,z }| −(1 − ϕ){(1 − λ)J0,z

µ,ϕ,η 1+µ,1+ϕ,1+η {f (z)} + λzJ0,z {f (z)} −α (1 − λ)J0,z ∞ =

k=n+1

(k−1)[1+λ(k−ϕ−1)]Γ(k+1)Γ(k−ϕ+η+1) k−1 a z k Γ(k−ϕ+1)Γ(k−µ+η+1)

∞ (1−λϕ)Γ(2−ϕ+η) − −α Γ(2−ϕ)Γ(k−µ+η)

k=n+1

≤

∞ k=n+1

[1+λ(k−ϕ−1)]Γ(k+1)Γ(k−ϕ+η+1) k−1 a z k Γ(k−ϕ+1)Γ(k−µ+η+1)

(k+α−1)[1+λ(k−ϕ−1)]Γ(k+1)Γ(k−ϕ+η+1) ak Γ(k−ϕ+1)Γ(k−µ+η+1)

−

α(1−λϕ)Γ(2−ϕ+η) Γ(2−ϕ)Γ(k−µ+η)

≤ 0 (n ∈ N ; 0 ≤ α ≤ 1; 0 ≤ µ < 1; ϕ, η ∈ R; ϕ < 1; η > max{µ, ϕ} − 2).

Hence, by maximum modulus theorem, the function f (z) defined by (1.1) belongs to the class Tλµ,ϕ,η (n; α).

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In order to prove the converse, we assume that the function f (z) ∈ Tλµ,ϕ,η (n; α). Then 1+µ,1+ϕ,1+η 2+µ,2+ϕ,2+η zJ {f (z)} + λz 2 J0,z {f (z)} 0,z − (1 − ϕ) µ,ϕ,η 1+µ,1+ϕ,1+η (1 − λ)J0,z {f (z)} + λzJ0,z {f (z)} ∞ (k−1)[1+λ(k−ϕ−1)]Γ(k+1)Γ(k−ϕ+η+1) k−1 a z k Γ(k−ϕ+1)Γ(k−µ+η+1) k=n+1 < α. (2.3) = ∞ [1+λ(k−ϕ−1)]Γ(k−ϕ+η+1) (k−1)[1+λ(k−ϕ−1)]Γ(k+1)Γ(k−ϕ+η+1) k−1 − a z Γ(k−ϕ+1)Γ(k−µ+η) k Γ(k−ϕ+1)Γ(k−µ+η+1) k=n+1

Since |(z)| ≤ |z| for any z , choosing z to be real and letting z → 1- through real values, (2.3) yields ∞ k=n+1

(k−1)[1+λ(k−ϕ−1)]Γ(k+1)Γ(k−ϕ+η+1) ak Γ(k−ϕ+1)Γ(k−µ+η+1)

≤α

(1−λϕ)Γ(2−ϕ+η) Γ(2−ϕ)Γ(k−µ+η)

−

∞ k=n+1

[1+λ(k−ϕ−1)]Γ(k+1)Γ(k−ϕ+η+1) ak Γ(k−ϕ+1)Γ(k−µ+η+1)

which is the desired assertion (2.1) of Theorem 1. Finally, by observing that the function f (z) given by (2.2) is indeed an extremal function for the assertion (2.1), we complete the proof of Theorem 1. Corollary 1.1. Let a function f (z) ∈ T (n). Then, the function f (z) belongs to the class V µ,ϕ,η (n; α) if and only if (2.4)

∞ αΓ(2 − ϕ + η) k!(k + α − 1)Γ(k − ϕ + η + 1) ak ≤ , Γ(k − ϕ + 1)Γ(k − µ + η + 1) Γ(2 − ϕ)Γ(2 − µ + η)

k=n+1

(n ∈ N ; 0 ≤ α ≤ 1; 0 ≤ µ < 1; ϕ, η ∈ R; ϕ < 2; η > max{µ, ϕ} − 2). Corollary 1.2. Let a function f (z) ∈ T (n). Then, the function f (z) belongs to the class W µ,ϕ,η (n; α) if and only if (2.5)

∞ αΓ(2 − ϕ + η) k!(k + α − 1)Γ(k − ϕ + η + 1) ak ≤ , Γ(k − ϕ)Γ(k − µ + η + 1) Γ(1 − ϕ)Γ(2 − µ + η)

k=n+1

(n ∈ N ; 0 ≤ α ≤ 1; 0 ≤ µ < 1; ϕ, η ∈ R; ϕ < 1; η > max{µ, ϕ} − 2). Corollary 1.3. Let a function f (z) ∈ T (n). Then, the function f (z) belongs to the class Ω µ (n; α) if and only if

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(2.6)

Hu¨ seyin Irmak and R. K. Raina ∞ α k!(k + α − 1) ak ≤ , (n ∈ N ; 0 < α ≤ 1; 0 ≤ µ < 1). Γ(k − µ + 1) Γ(2 − µ)

k=n+1

Corollary 1.4. Let a function f (z) ∈ T (n). Then, the function f (z) belongs to the class µ (n; α) if and only if (2.7)

∞ α k!(k + α − 1) ak ≤ , (n ∈ N ; 0 < α ≤ 1; 0 ≤ µ < 1). Γ(k − µ) Γ(1 − µ)

k=n+1

Corollary 1.5 [cf., [1] and [10]). Let a function f (z) ∈ T (n). Then, the function f (z) belongs to the class S n (α) if and only if ∞

(2.8)

(k + α − 1)ak ≤ α, (n ∈ N ; 0 < α ≤ 1).

k=n+1

Corollary 1.6 [cf., [1] and [10]). Let a function f (z) ∈ T (n). Then, the function f (z) belongs to the class C n (α) if and only if ∞

(2.9)

k(k + α − 1)ak ≤ α,

(n ∈ N ; 0 < α ≤ 1).

k=n+1

We next prove two distortion inequalities (Theorems 2 and 3 below) involving µ,β,η µ,β,η and J0,z , respectively. the fractional operators I0,z Theorem 2. Let β ∈ R+ and γ, η ∈ R such that η > max{−β, γ} − 2. If n is a positive integer such that (2.10)

n≥

γ(β + η) −2 β

and, if f (z) ∈ T λµ,ϕ,η (n; α), then β,γ,η I {f (z)} − 0,z (2.11)

≤

Γ(2 − γ + η) 1−γ |z| Γ(2 − γ)Γ(2 + β + η)

α(1 − λϕ)Γ(2 − ϕ + η)Γ(n − ϕ + 2)Γ(n − µ + η + 2) (n + α)Γ(2 − ϕ)Γ(2 − µ + η)Γ(n + β + η + 2)Γ(n − γ + 2) ·

Γ(n − γ + η + 2) |z|n−γ+1 Γ(n − ϕ + η + 2)[1 + λ(n − ϕ)]

for z ∈ U if γ ≤ 1 and z ∈ D if γ > 1. The result (2.11) is sharp for the function f (z) given by (2.2).


Proof.

Let f (z) ∈ T λµ,ϕ,η (n; α). It follows from the inequality (2.1) that ∞

(2.12)

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ak ≤

k=n+1

α(1 − λϕ)Γ(2 − ϕ + η)Γ(n − ϕ + 2) (n + α)[1 + λ(n − ϕ)]Γ(2 − ϕ)Γ(2 − µ + η) ·

Γ(n − µ + η + 2) . Γ(n + 2)Γ(n − ϕ + η + 2)

From (1.1), (1.8), and a known result due to Srivastava et al. [10, p. 415, Eq. (2.3)], we have ∞ Γ(2 − γ + η) β,γ,η 1−γ (2.13) I0,z {f (z)} = z − Ψ(k)ak z k−γ , Γ(2 − γ)Γ(2 + β + η) k=n+1

where Ψ(k) =

Γ(k + 1)Γ(k − γ + η + 1) , (k ≥ n + 1; n ∈ N ). Γ(k − γ + 1)Γ(k + β + η + 1)

The function Ψ(z) is non-increasing for integers k (k ≥ n + 1; n ∈ N ), under the hypotheses of Theorem 2 including the constraint (2.10). Therefore, we have (2.14)

0 < Ψ(k) ≤ Ψ(n + 1) =

Γ(n + 2)Γ(n − γ + η + 2) , (n ∈ N ). Γ(n − γ + 2)Γ(n + β + η + 2)

The the desired assertion (2.11) of the theorem follows on combining the two inequalities which straigthforwardly emerge from (2.13) upon suitably using (2.12) and (2.14) in the process. Theorem 3. Let 0 ≤ β < 1 and γ, η ∈ R such that γ < 2, η > max{β, γ} − 2. If n is a positive integer such that (2.15)

n≥

γ(β − η) − 2, β

and, if f (z) ∈ T λµ,ϕ,η (n; α), then β,γ,η Γ(2 − γ + η) 1−γ J 0,z {f (z)} − Γ(2 − γ)Γ(2 − β + η) |z| (2.16)

≤

α(1 − λϕ)Γ(2 − ϕ + η)Γ(n − ϕ + 2)Γ(n − µ + η + 2) (n + α)Γ(2 − ϕ)Γ(2 − µ + η)Γ(n − β + η + 2)Γ(n − γ + 2) ·

Γ(n − γ + η + 2) |z|n−γ+1 Γ(n − ϕ + η + 2)[1 + λ(n − ϕ)]

for z ∈ U if γ ≤ 1 and z ∈ D if γ > 1. The result is sharp for the function f (z) given by (2.2).

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Proof. Under the hypotheses of Theorem 3, we have from (1.1) and a known result due to Raina and Srivastava [7, p. 15, Eq. (2.2)]: β,γ,η

J0,z {f (z)} = (2.17) −

∞

k=n+1

Γ(2 − γ + η) z 1−γ Γ(2 − γ)Γ(2 − β + η) Γ(k + 1)Γ(k − γ + η + 1) z k−γ . Γ(k − γ + 1)Γ(k − β + η + 1)

Making use of (2.17), and the following steps similar to those given in the proof of Theorem 2, the assertion (2.16) of Theorem 3 is easily arrived at. A number of simpler distortion properties can easily be deduced from Theorems 2 and 3 by suitably specializing the parameters γ, ϕ, λ, α, and n in the equations (2.11) and (2.16). For γ = −β in Theorem 2 and γ = β in Theorem 3, we get following corollaries. Corollary 1.7. If f (z) ∈ T λµ,ϕ,η (n; α), then α(1−λϕ)Γ(2−ϕ+η)Γ(n−ϕ+2) −β 1 1+β Dz {f (z)}− |z| ≤ (n + α)Γ(2 − ϕ)Γ(2 − µ + η) Γ(2 + β) (2.18) Γ(n − µ + η + 2) |z|n+β+1 · Γ(n + β + 2)Γ(n − ϕ + η + 2)[1 + λ(n − ϕ)] for all β (β > 0), z ∈ U and n ∈ N . µ,ϕ,η

(n; α), then Corollary 1.8. If f (z) ∈ T λ α(1−λϕ)Γ(2− ϕ+η)Γ(n−ϕ+2) β 1 1−β Dz {f (z)}− |z| ≤ Γ(2−β) (n + α)Γ(2−ϕ)Γ(2−µ+η) (2.19) Γ(n − µ + η + 2) |z|n−β+1 · Γ(n − β + 2)Γ(n − ϕ + η + 2)[1 + λ(n − ϕ)] for all β (0 ≤ β < 1), z ∈ U and n ∈ N . Each of these results in (2.18) and (2.19) is sharp for the function f (z) given by f (z) = z − (2.20) ·

α(1 − λϕ)Γ(2 − ϕ + η)Γ(n − ϕ + 2) (n + α)[1 + λ(n − ϕ)]Γ(2 − ϕ)Γ(2 − µ + η)

Γ(n − µ + η + 2) z n+1 , (n ∈ N ). Γ(n + 2)Γ(n − ϕ + η + 2)

Finally, Theorem 4 below gives the radii of starlikeness, convexity, and closeto-convexity of functions f (z) belonging to the class Tλµ,ϕ,η (n; α). These results are proven by appropriately using the inequalities (1.2)-(1.4) in the forms:

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(2.21)

zf (z) f (z) − 1 < 1 − δ, (0 ≤ δ < 1; 0 < |z| < Γ1 ), zf (z) f (z) < 1 − δ,

(2.22)

(0 ≤ δ < 1; 0 < |z| < Γ2 ),

|f (z) − 1| < 1 − δ, (0 ≤ δ < 1; 0 < |z| < Γ3 ),

(2.23)

respectively, and we omit the details involved, r1 , r2 , and r3 being defined by (2.25)-(2.27). Theorem 4.

µ,ϕ,η

If f (z) ∈ Tλ

(n; α). Then, the function f (z) is

(i) starlike of order δ(0 ≤ δ < 1) in 0 < |z| < r 1 , (ii) convex of order δ(0 ≤ δ < 1) in 0 < |z| < r 2 , (iii) close-to-convex of order δ(0 ≤ δ < 1) in 0 < |z| < r 3 , where 1

(2.25) r1 = r1 (n; α; δ; λ; µ; ϕ; η) = inf [(1 − δ) · Ω(k, n, λ, α, µ, η, ϕ)] k−1 , k≥n+1

(2.26) r2 = r2 (n; α; δ; λ; µ; ϕ; η) = inf

k≥n+1

(2.27)

r3 = r3 (n; α; δ; λ; µ; ϕ; η) = inf

k≥n+1

k(1−δ) · Ω(k, n, λ, α, µ, η, ϕ) k−δ

1−δ · Ω(k, n, λ, α, µ, η, ϕ) k−δ

1 k−1

1 k−1

,

,

f or all n ∈ N , when Ω(k, n, λ, α, µ, η, ϕ) =

[1 + λ(n − ϕ)]Γ(k)Γ(2 − ϕ)Γ(2 − µ + η)Γ(2 − ϕ + η) . α(1 − λϕ)Γ(2 − ϕ + η)Γ(k − µ + η + 1)

We conclude by remarking that several results of interest giving the coefficient bounds, distortion inequalities, radii of close-to-convexity, starlikeness and convexity of functions which belong to various subclasses of T (n) can be obtained by suitable choices of the involved parameters from the results presented in this paper. We skip further details in this regard. ACKNOWLEDGEMENT ˙ ¨ ITAK This paper was supported, in part, by TUB (The Scientific end Technical Research Council of Turkey) and Bas¸kent University (Ankara, Turkey). The author

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would also like to acknowledge Professor Dr. Mehmet Haberal, Rector of Bas¸kent University, who generously supports scientific researches in all aspects. The authors express their sincerest thanks to the referee for valuable suggestions. The secondnamed author would like to express his thanks, to the council for Scientific and Industrial Research (Govt. of India) for financial assistance. REFERENCES 1. O. Altintas¸, H. Irmak, and H. M. Srivastava, A subclass of analytic functions defined by using certain operators of fractional calculus, Comput. Math. Appl. 30 (1995), 1-9. 2. O. Altintas¸, H. Irmak, and H. M. Srivastava, Fractional calculus and certain starlike functions with negative coefficients, Comput. Math. Appl. 30 (1995), 9-15. 3. M.-P. Chen, H. Irmak, and H. M. Srivastava, Some families of multivalently analytic functions with negative coefficients, J. Math. Anal. Appl. 214 (1997), 674-690. 4. M.-P. Chen, H. Irmak, and H. M. Srivastava, A certain subclass of analytic functions involving operators of fractional calculus, Comput. Math. Appl. 35 (1998), 81-91. 5. P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Bd. 259, Springer-Verlag, Berlin, Heidelberg, and Tokyo, 1983. 6. H. Irmak, Certain subclasses of p - valently starlike functions with negative coefficients, Bull. Calcutta Math. Soc. 87 (1995), 589-598. 7. R. K. Raina, and H. M. Srivastava, A certain subclass of analytic functions associated with operators of fractional calculus, Comput. Math. Appl. 32 (1996), 13-19. 8. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Reading, PA, 1993. 9. H. M. Srivastava, and S. Owa, (Editors), Current Topics in Analytic Function Theory, World Scientific, Singapore, 1992. 10. H. M. Srivastava, M. Saigo, and S. Owa, A class of distortion theorems involving certain operators of fractional calculus, J. Math. Anal. Appl., 131 (1988), 412-420. Hu¨ seyin Irmak Department of Mathematics Education, Faculty of Education, Bas¸kent University, Ba ˘ glica Campus, Ba ˘ glica 06530-Etimesgut, Ankara, Turkey E-mail: [email protected] R. K. Raina Department of Mathematics, C.T.A.E. M.P. University of Agri. & Technology, Udaipur 313001, Rajasthan, India E-mail: rainark− [email protected]