SUBORDINATION AND SUPERORDINATION FOR FUNCTIONS BASED ON DZIOK-SRIVASTAVA LINEAR OPERATOR

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1 Bulletin of Mathematical Analysis and Applications ISSN: , URL: Volume 2, Issue 3(21), Pages SUBORDINATION AND SUPERORDINATION FOR FUNCTIONS BASED ON DZIOK-SRIVASTAVA LINEAR OPERATOR RABHA W. IBRAHIM AND MASLINA DARUS Abstract. In this article, we obtain some subordination and superordination results involving Dziok-Srivastava linear operator and fractional integral operator for certain normalized analytic functions in the open unit disk. 1. Introduction and Preliminaries. Let H(U) denote the class of analytic functions in the unit disk For n positive integer and a C, let U := {z C, z < 1}. H[a, n] := {f H(U) : f(z) = a + a n z n + a n+1 z n , z U}, and A n = {f H(U) : f(z) = z + a n z n + a n+1 z n , z U} with A 1 = A. A function f H[a, n] is convex in U if it is univalent and f(u) is convex. It is well-known that f is convex if and only if f() = and R{1 + zf (z) f } >, z U. (z) Definition 1.1. [1] Denote by Q the set of all functions f(z) that are analytic and injective on U E(f) where E(f) := {ζ U : lim z ζ f(z) = } and are such that f (ζ) = for ζ U E(f). Given two functions F and G in the unit disk U, the function F is subordinate to G, written F G, if G is univalent, F () = G() and F (U) G(U). Alternatively, given two functions F and G, which are analytic in U, the function F is said to be subordinate to G in U if there exists a function h, analytic in U with such that h() = and h(z) < 1 for all z U F (z) = G(h(z)) for all z U. 2 Mathematics Subject Classification. 34G1, 26A33,3C45. Key words and phrases. Fractional calculus; Univalent solution; Subordination; Superordination. c 21 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted May 6, 21. Published June 9, 21. The authors were partially supported by UKM-ST-6-FRGS17-29, MOHE Malaysia. 15

2 16 R. W. IBRAHIM, M. DARUS Let φ : C 2 C and let h be univalent in U. If p is analytic in U and satisfies the differential subordination φ(p(z)), zp (z)) h(z), then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, if p q. If p and φ(p(z)), zp (z)) are univalent in U and satisfy the differential superordination h(z) φ(p(z)), zp (z)), then p is called a solution of the differential superordination. An analytic function q is called subordinant of the solution of the differential superordination if q p. We shall need the following results: Lemma 1.1. [2] Let q be univalent in the unit disk U, and let θ and φ be analytic in a domain D containing q(u) with φ(w) = when w q(u). Set Q(z) := zq (z)φ(q(z)), h(z) := θ(q(z)) + Q(z). Suppose that 1. Q(z) is starlike univalent in U, and 2. R zh (z) Q(z) > for z U. If θ(p(z)) + zp (z)φ(p(z)) θ(q(z)) + zq (z)φ(q(z)), then p(z) q(z) and q is the best dominant. Lemma 1.2. [3] Let q be convex univalent in the unit disk U and ψ and γ C with R{1+ zq (z) q (z) + ψ γ } >. If p(z) is analytic in U and ψp(z)+γzp (z) ψq(z)+γzq (z), then p(z) q(z) and q is the best dominant. Lemma 1.3. [4] Let q be convex univalent in the unit disk U and θ and φ be analytic in a domain D containing q(u). Suppose that 1. zq (z)φ(q(z)) is starlike univalent in U, and 2. R{ θ (q(z)) φ(q(z)) } > for z U. If p(z) H[q(), 1] Q, with p(u) D and θ(p(z)) + zp (z)φ(z) is univalent in U and θ(q(z)) + zq (z)φ(q(z)) θ(p(z)) + zp (z)φ(p(z)) then q(z) p(z) and q is the best subordinant. Lemma 1.4. [1] Let q be convex univalent in the unit disk U and γ C. Further, assume that R{γ} >. If p(z) H[q(), 1] Q, with p(z) + γzp (z) is univalent in U then q(z) + γzq (z) p(z) + γzp (z) implies q(z) p(z) and q is the best subordinant. For two functions f(z) = z + n=2 a nz n and g(z) = z + n=2 b nz n, the Hadamard product (or convolution) of f and g defined by (f g)(z) := z + a n b n z n =: (g f)(z). n=2 For α j C (j = 1, 2,..., l) and β j C {, 1, 2,...} (j = 1, 2,..., m), the generalized hypergeometric function l F m (α 1,..., α l ; β 1,...β m ; z) is defined by the infinite series (α 1 ) n...(α l ) n z n lf m (α 1,..., α l ; β 1,..., β m ; z) := (β 1 ) n...(β m ) n n! n= (l m + 1 : l, m N := {, 1, 2,...}) where (a) n is the Pochhammer symbol defined by { Γ(a + n) 1, (n = ); (a) n := = Γ(a) a(a + 1)(a + 2)...(a + n 1), (n N).

3 SUBORDINATION AND SUPERORDINATION 17 Corresponding to the function h(α 1,..., α l ; β 1,...β m ; z) := z l F m (α 1,..., α l ; β 1,...β m ; z), the Dziok-Srivastava operator (see [5-7]) H l m(α 1,..., α l ; β 1,...β m ) is defined by the Hadamard product Hm(α l 1,..., α l ; β 1,...β m )f(z) := h(α 1,..., α l ; β 1,...β m ; z) f(z) (α 1 ) n 1...(α l ) n 1 a n z n = z + (β 1 ) n 1...(β m ) n 1 (n 1)! We can verify that n=2 := H l m[α 1 ]f(z). z(h l m[α 1 ]f(z)) = α 1 H l m[α 1 + 1]f(z) (α 1 1)H l m[α 1 ]f(z). Special cases of the Dziok-Srivastava linear operator include the Hohlov linear operator [8], the Carlson-Shaffer linear operator L(a, c) [9], the Ruscheweyh derivative operator D n [1], the generalized Bernardi-Libera-Livingston linear integral operator [11] and the Srivastava-Owa fractional derivative operator [12]: Definition 1.2. The fractional derivative of order α is defined, for a function f by Dz α 1 d f(z) := Γ(1 α) dz f(ζ) dζ; α < 1, (z ζ) α where the function f is analytic in simply-connected region of the complex z-plane C containing the origin and the multiplicity of (z ζ) α is removed by requiring log(z ζ) to be real when(z ζ) >. Definition 1.3. The fractional integral of order α is defined, for a function f, by I α z f(z) := 1 f(ζ)(z ζ) α 1 dζ; α >, where the function f is analytic in simply-connected region of the complex z-plane (C) containing the origin and the multiplicity of (z ζ) α 1 is removed by requiring log(z ζ) to be real when(z ζ) >. Remark 1.1. [12] D α z {z μ } = Γ(μ + 1) Γ(μ α + 1) {zμ α }, μ > 1; α < 1 and Iz α {z μ Γ(μ + 1) } = Γ(μ + α + 1) {zμ+α }, μ > 1; α >. The main object of the present paper is to find the sufficient conditions for certain normalized analytic functions f, g to satisfy and [ Iα z H l m[α 1 ]g 1 (z) ] μ [ Iα z H l m[α 1 ]f(z) ] μ [ Iα z Hm[α l 1 ]g 2 (z) ] μ q 1 (z) [ Iα z Hm[α l 1 ]f(z) ] μ q 2 (z), =, z U

4 18 R. W. IBRAHIM, M. DARUS where μ 1, q 1 and q 2 are given univalent functions in U. Also, we obtain the results as special cases. Further, in this paper, we study the existence of univalent solution for the fractional differential equation D α z u(z) = H l m[α 1 ]f(z), (1.1) subject to the initial condition u() =, where u : U C is an analytic function for all z U, ρ : U C {} is an analytic functions in z U and f : U C is a univalent function in U. The existence is obtained by applying Schauder fixed point theorem. Moreover, we discuss some properties of this solution involving fractional differential subordination. The following results are used in the sequel. Theorem 1.1. (Arzela-Ascoli) (see [13]) Let E be a compact metric space and C(E) be the Banach space of real or complex valued continuous functions normed by f := sup t E f(t). If A = {f n } is a sequence in C(E) such that f n is uniformly bounded and equicontinuous, then A is compact. Let M be a subset of Banach space X and A : M M an operator. The operator A is called compact on the set M if it carries every bounded subset of M into a compact set. If A is continuous on M (that is, it maps bounded sets into bounded sets ) then it is said to be completely continuous on M. Theorem 1.2. (Schauder) (see [14]) Let X be a Banach space, M X a nonempty closed bounded convex subset and P : M M is compact. Then P has a fixed point. Recently, the subordination and superordination containing the Dziok-Srivastava linear operator are studied by many authors [15]. 2. Subordination and superordination. In this section, we study some important properties of the fractional differential and integral operators D α z, I α z, given by the authors [16] which are useful in the next results of the subordination and superordination. Theorem 2.1[16] For α, (, 1] and f is a continuous function, then 1 DI α z f(z) = (z)α 1 f() + Iα z Df(z); D = d dz. 2 I α z D α z f(z) = D α z I α z f(z) = f(z). But, first we consider the subordination results involving Dziok-Srivastava linear operator and fractional integral operator as the following: Theorem 2.2. Let f, g be analytic in U. [ Iα z Hl m [α1]g(z) ] μ be univalent in U such that Iα z Hl m [α1]g(z) subordination = and z([ Iα z Hl m [α1]g(z) ] μ ) be starlike univalent in U. If the [ Iα z Hm[α l 1 ]f(z) ] μ {1 + μ( ziα [Hm[α l 1 ]f(z)]

5 holds and where Then SUBORDINATION AND SUPERORDINATION 19 [ Iα z Hm[α l 1 ]g(z) ] μ {1 + μ( ziα [Hm[α l 1 ]g(z)] 1 ]g(z) ρ(z) } R{ zg (z) G(z) + (μ 1) zg(z) Iz α Hm[α l } >, z U, 1 ]g(z) G(z) := [ Iα z Hm[α l 1 ]g(z) ]. [ Iα z H l m[α 1 ]f(z) and [ Iα z Hl m [α1]g ] μ is the best dominant. Proof. Setting p(z) := [ Iα z H l m[α 1 ]f(z) ] μ [ Iα z Hm[α l 1 ]g(z) ] μ ] μ, q(z) := [ Iα z Hm[α l 1 ]g(z) ] μ. Our aim is to apply Lemma 1.1. First we show that R{1 + zq (z) q (z) } >. Assume that R{1 + zq (z) q (z) } = R{1 + zg (z) G(z) θ(ω) := ω and φ(ω) := 1, G(z) + (μ 1) Iz α Hm[α l 1 ]g(z) } >. it can easily be observed that θ, φ are analytic in C. Also, we let Q(z) := zq (z)φ(z) = zq (z), h(z) := θ(q(z)) + Q(z) = q(z) + zq (z). By the assumptions of the theorem we find that Q is starlike univalent in U and that R{ zh (z) Q(z) } = R{2 + zq (z) q (z) } >. By using Theorem 2.1, a computation shows p(z) + zp (z) = [ Iα z Hm[α l 1 ]f(z) ] μ {1 + μ( ziα [Hm[α l 1 ]f(z)] [ Iα z H l m[α 1 ]g(z) = q(z) + zq (z). ] μ {1 + μ( ziα [Hm[α l 1 ]g(z)] Iz α Hm[α l ) zρ (z) 1 ]g(z) ρ(z) )} Thus in view of Lemma 1.1, p(z) q(z) and q is the best dominant. Corollary 2.1. Let f, g be analytic in U. [ Iα z L(a,c)g ] μ be univalent in U and z([ Iα z L(a,c)g ] μ ) be starlike univalent in U. If the subordination [ Iα z L(a, c)f(z) ] μ {1 + μ( ziα [L(a, c)f(z)] Iz α zρ (z) L(a, c)f(z) ρ(z) )} [ Iα z L(a, c)g(z) ] μ {1 + μ( ziα [L(a, c)g(z)] Iz α zρ (z) L(a, c)g(z) ρ(z) )}

6 2 R. W. IBRAHIM, M. DARUS holds and where Then R{ zg (z) G(z) + (μ 1) zg(z) Iz α } >, z U, L(a, c)g(z) G(z) := [ Iα z L(a, c)g(z) ]. [ Iα z L(a, c)f(z) and [ Iα z L(a,c)g ] μ is the best dominant. ] μ [ Iα z L(a, c)g(z) ] μ Proof. By putting l = 2, m = 1, α 1 = a, α 2 = 1 and β 1 = c in Theorem 2.2. Corollary 2.2. I α z Hl m [α1]g(z) Let f, g be analytic in U, [ Iα z g ] μ be univalent in U such that = and z([ Iα z g ] μ ) be starlike univalent in U. If the subordination [ Iα z f(z) ]μ {1 + μ( ziα [f(z)] Iz α zρ (z) f(z) ρ(z) )} [Iα z g(z) ]μ {1 + μ( ziα [g(z)] Iz α zρ (z) g(z) ρ(z) )} holds and Then R{ zg (z) G(z) + (μ 1)zG(z) Iz α } >, z U, where G(z) := [ Iα z g(z) g(z) ]. and [ Iα z g ] μ is the best dominant. [ Iα z f(z) ]μ [ Iα z g(z) ]μ Proof. By putting l = 1, m =, α 1 = 1, in Theorem 2.2. Theorem 2.3. Let f, g be analytic in U, q be convex univalent in U with R{1 + zq (z) q (z) + 1 γ }, γ C and [ Iα z Hl m [α1]f ] μ be analytic in U. If the subordination [ Iα z Hm[α l 1 ]f(z) ] μ {1 + μγ( ziα [Hm[α l 1 ]f(z)] q(z) + γzq (z) holds. Then and q is the best dominant. Proof. Setting [ Iα z Hm[α l 1 ]f(z) ] μ q(z) p(z) := [ Iα z Hm[α l 1 ]f(z) ] μ. Our aim is to applied Lemma 1.2. Let ψ := 1, since p(z) + γzp (z) = [ Iα z H l m[α 1 ]f(z) ] μ + γz([ Iα z Hm[α l 1 ]f(z) ] μ ) = [ Iα z Hm[α l 1 ]f(z) ] μ {1 + μγ( ziα [Hm[α l 1 ]f(z)] q(z) + γzq (z)

7 SUBORDINATION AND SUPERORDINATION 21 then, in view of Lemma 1.2, p(z) q(z) and q is the best dominant. Corollary 2.3. Let f, g be analytic in U, 1 B A 1, q(z) := [ 1+Az 1+Bz ]μ with R{1 + zq (z) q (z) [ Iα z H l m[α 1 ]f(z) holds. Then + 1 γ }, γ C and [ Iα z Hl m [α1]f ] μ be analytic in U. If the subordination ] μ {1+μ( ziα z [H l m[α 1 ]f(z)] I α z H l m[α 1 ]f(z) zρ (z) 1 + Az )} [ ρ(z) [ Iα z Hm[α l 1 ]f(z) ] μ [ 1 + Az 1 + Bz ]μ, 1 B < A 1 and [ 1+Az 1+Bz ]μ is the best dominant. μγz(a B) 1 + Bz ]μ {1+ (1 + Az)(1 + Bz) } Next, applying Lemma 1.3 and Lemma 1.4 respectively, to obtain the following theorems. Theorem 2.4. U such that Iα z Hl m [α1]g(z) Let f, g be analytic in U,[ Iα z Hl m [α1]g ] μ be convex univalent in =, z([ Iα z Hl m [α1]g ] μ ) be starlike univalent in U and (z[ Iα z Hl m [α1]f ] μ ) be univalent in U. If the subordination holds and [ zα 1 [ Iα z Hm[α l 1 ]g(z) ] μ {1 + μ( ziα [Hm[α l 1 ]g(z)] 1 ]g(z) ρ(z) )} [ Iα z Hm[α l 1 ]f(z) ] μ {1 + μ( ziα [Hm[α l 1 ]f(z)] ]μ [ Hl m [α1]f(z) ] μ H[, 1] Q. Then [ Iα z H l m[α 1 ]g(z) and [ Iα z Hl m [α1]g ] μ is the best subordinant. Proof. Setting p(z) := [ Iα z H l m[α 1 ]f(z) Our aim is to apply Lemma 1.3. By taking ] μ [ Iα z Hm[α l 1 ]f(z) ] μ ] μ, q(z) := [ Iα z Hm[α l 1 ]g(z) ] μ. θ(ω) := ω and φ(ω) := 1, it can easily observed that θ, φ are analytic in C. Thus Now we must show that R{ θ (q(z)) φ(q(z)) } = 1 >. q(z) + zq (z) p(z) + zp (z).

8 22 R. W. IBRAHIM, M. DARUS a computation shows that q(z) + zq (z) = [ Iα z Hm[α l 1 ]g(z) ] μ {1 + μ( ziα [Hm[α l 1 ]g(z)] 1 ]g(z) ρ(z) )} [ Iα z Hm[α l 1 ]f(z) ] μ {1 + μ( ziα [Hm[α l 1 ]f(z)] = p(z) + zp (z). Thus in view of Lemma 1.3, q(z) p(z) and p is the best subordinant. Theorem 2.5. Let f, g be analytic in U, q be convex univalent in U, [ Iα z Hl m [α1]f(z) ] μ H[, 1] Q and [ Iα z Hm[α l 1 ]f(z) ] μ {1 + μγ( ziα [Hm[α l 1 ]f(z)] Iz α Hm[α l 1 ]f(z) be univalent in U. If the subordination zρ (z) )}, R{γ} >, ρ(z) q(z) + γzq (z) [ Iα z Hm[α l 1 ]f(z) ] μ {1 + μγ( ziα [Hm[α l 1 ]f(z)] holds. Then q(z) [ Iα z Hm[α l 1 ]f(z) and q is the best subordinant. Proof. Setting p(z) := [ Iα z Hm[α l 1 ]f(z) ] μ. Our aim is to apply Lemma 1.4. Since q(z) + γzq (z) = [ Iα z Hm[α l 1 ]g(z) ] μ {1 + μγ( ziα [Hm[α l 1 ]g(z)] 1 ]g(z) ρ(z) )} [ Iα z Hm[α l 1 ]f(z) ] μ {1 + μγ( ziα [Hm[α l 1 ]f(z)] = p(z) + γzp (z) then, in view of Lemma 1.4, q(z) p(z) and q is the best subordinant. Combining the results of differential subordination and superordination, we state the following sandwich theorems. Theorem 2.6. Let f, g 1, g 2 be analytic in U, [ Iα z Hl m [α1]g1 ] μ be convex univalent in U such that Iα z Hl m [α1]g(z) in U and let (z[ Iα z Hl m [α1]f ] μ =, z([ Iα z Hl m [α1]g1 ] μ ), z([ Iα z Hl m [α1]g2 ] μ ) be starlike univalent ] μ ), [ Iα z Hl m [α1]g2 ] μ be univalent in U. If the subordination [ Iα z Hm[α l 1 ]g 1 (z) ] μ {1 + μ( ziα [Hm[α l 1 ]g 1 (z)] 1 ]g 1 (z) ρ(z) )} [ Iα z Hm[α l 1 ]f(z) ] μ {1 + μ( ziα [Hm[α l 1 ]f(z)]

9 holds, [ zα 1 where Then SUBORDINATION AND SUPERORDINATION 23 [ Iα z Hm[α l 1 ]g 2 (z) ] μ {1 + μ( ziα [Hm[α l 1 ]g 2 (z)] 1 ]g 2 (z) ρ(z) )} ]μ [ Hl m [α1]f(z) ] μ H[, 1] Q and R{ zg 2(z) G 2 (z) + (μ 1) zg 2(z) Iz α Hm[α l } >, z U, 1 ]g 2 (z) [ Iα z H l m[α 1 ]g 1 (z) G 2 (z) := [ Iα z Hm[α l 1 ]g 2 (z) ]. ] μ [ Iα z H l m[α 1 ]f(z) ] μ [ Iα z Hm[α l 1 ]g 2 (z) ] μ and [ Iα z Hl m [α1]g1 ] μ and [ Iα z Hl m [α1]g2 ] μ are respectively the best subordinant and dominant. Theorem 2.7. Let f, g 1, g 2 A, q 1, q 2 be convex univalent in U, with R{1 + zq 2 (z) q 2 (z) + 1 γ }, γ C, [ Iα z Hl m [α1]f ] μ H[, 1] Q, and analytic in U and [ Iα z Hm[α l 1 ]f ] μ {1 + μ( ziα [Hm[α l 1 ]f] Iz α Hm[α l 1 ]f be univalent in U. If the subordination q 1 (z)+γzq 1(z) [ Iα z H l m[α 1 ]f(z) holds. Then zρ )}, R{γ} >, ρ ] μ {1+μγ( ziα [Hm[α l 1 ]f(z)] q 2(z)+γzq 2(z) q 1 (z) [ Iα z Hm[α l 1 ]f(z) ] μ q 2 (z) and q 1, q 2 are respectively the best subordinant and the best dominant. 3. Existence of univalent solution. Let B := C[U, C] be a Banach space of all continuous functions on U endowed with the sup. norm u := sup z U u(z). By using the properties in Theorem 2.1, we can easily obtain the following result: Lemma 3.1. If the function f A, then the initial value problem (1.1) is equivalent to the nonlinear integral equation u(z) = 1 (z ζ) α 1 H l m[α 1 ]f(ζ)dζ. (3.1) In other words, every solution of the equation (3.1) is also a solution of the initial value problem (1.1) and vice versa. 1 Theorem 3.1.(Existence) Assume that M; M >. Then there exists a univalent function u : U C solving the problem (1.1). Proof. Define an operator P : C C (P u)(z) := 1 (z ζ) α 1 H l m[α 1 ]f(ζ)dζ. (3.2)

10 24 R. W. IBRAHIM, M. DARUS Denotes B n := (α1)n 1...(α l) n 1 1 (β 1) n 1...(β m) n 1 (n 1)!. Our aim is to apply Theorem 2.1. First we show that P is bounded operator: Thus we obtain that 1 (P u)(z) = (z ζ) α 1 H l m[α 1 ]f(ζ)dζ 1 (z ζ) α 1 H l m[α 1 ]f(ζ)dζ (z ζ) α 1 < M(1 + B n a n ) dζ n=2 z α = M(1 + B n a n ) Γ(α + 1) n=2 < M(1 + n=2 B n a n ) Γ(α + 1) P < M(1 + n=2 B n a n ) := r Γ(α + 1) that is P : B r B r. Then P maps B r into itself. Now we proceed to prove that P is equicontinuous. For z 1, z 2 U such that z 1 = z 2, z 2 z 1 < δ, δ > Then for all u S, where S := {u C, : u M(1 + n=2 B n a n ) := r, r > }, Γ(α + 1) we obtain (P u)(z 1 ) (P u)(z 2 ) M(1 + n=2 B n a n ) 1 (z 1 ζ) α 1 dζ 2 (z 2 ζ) α 1 dζ M(1 + n=2 B n a n ) 1 [(z 1 ζ) α 1 (z 2 ζ) α 1 ] dζ + 2 z 1 = M(1+ n=2 Bn an ) Γ(α+1) [2(z 2 z 1 ) α + z α 2 z α 1 ] < 2M(1+ n=2 Bn an ) Γ(α+1) < 2M(1+ n=2 Bn an ) Γ(α+1) δ α, z 2 z 1 α (z 2 ζ) α 1 dζ which is independent on u. Hence P is an equicontinuous mapping on S. By the assumption of the theorem we can show that P is a univalent function (see [17]). The Arzela-Ascoli theorem yields that every sequence of functions from P (S) has got a uniformly convergent subsequence, and therefore P (S) is relatively compact. Schauder, s fixed point theorem asserts that P has a fixed point. By construction, a fixed point of P is a univalent solution of the initial value problem (1.1). The next theorems show the relation between univalent solutions and the subordination for a class of fractional differential problem. Theorem 3.2. Let the assumptions of Theorem 2.6 be satisfied. Then univalent solutions u 1, u, u 2, of the problem D α z u(z) = F (z, u(z)), (3.3)

11 SUBORDINATION AND SUPERORDINATION 25 subject to the initial condition u() =, where u : U C is an analytic function for all z U and F : U C C, is an analytic functions in z U, are satisfying the subordination u 1 u u 2. Proof. Setting μ = 1 and let F (z, u 1 (z)) := Hl m [α1]g1(z), F (z, u(z)) := Hl m [α1]f(z), and F (z, u 2 (z)) := Hl m [α1]g2(z) where =, z U. Theorem 3.3. Let the assumptions of Theorem 2.7 be satisfied. Then every univalent solution u(z) of the problem (3.3) satisfies the subordination q 1 (z) u(z) q 2 (z), where q 1 (z) and q 2 (z) are univalent function in U. Proof. Setting μ = 1, F (z, u(z)) := Hl m [α1]f(z). Acknowledgement: The authors would like to thank the anonymous referee for the informative and creative comments given to the article. References [1] S.S.Miller and P.T.Mocanu, Subordinants of differential superordinations, Complex Variables, 48(1)(23), [2] S.S.Miller and P.T.Mocanu, Differential Subordinantions: Theory and Applications. Pure and Applied Mathematics No.225 Dekker, New York, (2). [3] T.N.Shanmugam and V.Ravichangran and S.Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions, Austral. J. Math, Anal.Appl.3(1)(26),1-11. [4] T.Bulboaca, Classes of first-order differential superordinations, Demonstr.Math. 35(2)(22), [5] J.Dziok and H.M.Srivastava, Certain subclasses of analytic functons associated with the generalized hypergeometric function, Appl. Math. Comput., 13(1)(1999),1-13. [6] J.Dziok and H.M.Srivastava, Certain subclasses of analytic functons associated with the generalized hypergeometric function, Integral Transforms Spec. Funct., 14(1).,(23), [7] H.M.Srivastava, Some families of fractional derivative and other linear operators associated with analytic, univalent and multivalent functions, Proc. International Conf. Analysis and its Applications, Allied Publishers Ltd, New Delhi (21), [8] JU.E.Hohlov, Operators and operations on the class of univalent functions, Izv, Vyssh. Uchebn. Zaved. Math., 1(197)(1978) [9] BC.Carlson and D.B.Shaffer, Starlike and prestarlike hypergeometric functions, SIAM. J. Math. Anal., 15(4)(1984), [1] S.Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), [11] S.D.Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135(1969), [12] H.M.Srivastava and S.Owa, Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, [13] R.F.Curtain and A.J.Pritchard, Functional Analysis in Modren Applied Mathematics, Academic Press [14] K.Balachandar and J.P.Dauer, Elements of Control Theory, Narosa Publishing House,1999. [15] S.P. Goyal and R. Kumar, Subordination and Superordination Results of Non-Bazilevic Functions Involving Dziok-Srivastava Operator, Int. J. Open Problems Complex Analysis, Vol. 2, No. 1, March (21), [16] R.W.Ibrahim and M.Darus, Subordination and superordination for univalent solutions for fractional differential equations, J. Math. Anal. Appl. 345(28), [17] A.W.Goodman, Univalent Function, Mariner Publishing Company, INC, 1983.

12 26 R. W. IBRAHIM, M. DARUS Rabha W. Ibrahim and Maslina Darus School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi 436 Selangor D.E., Malaysia address: address: (corresponding author)