(1.1) Also, letA(n) be the subclass of the functionsf ∈H(U) of the form f(z) =z+ P∞ k=n+1 akzk, (1.2) and setA≡A(1)

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September 2011

ON SANDWICH THEOREMS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING EXTENDED

MULTIPLIER TRANSFORMATIONS

M. K. Aouf, A. Shamandy, R. M. El-Ashwah and E. E. Ali

Abstract. In this paper we derive some subordination and superordination results for certain normalized analytic functions in the open unit disc, which are acted upon by a class of extended multiplier transformations. Relevant connections of the results, which are presented in this paper, with various known results are also considered.

1. Introduction

LetH(U) be the class of analytic functions in the open unit discU ={z∈C:

|z|<1} and letH[a, n] denote the subclass of the functionsf ∈H(U) of the form f(z) =a+anzⁿ+an+1zⁿ⁺¹+· · · (a∈C). (1.1) Also, letA(n) be the subclass of the functionsf ∈H(U) of the form

f(z) =z+ P^∞

k=n+1

akz^k, (1.2)

and setA≡A(1).

Forf, g∈H(U), we say that the function f(z) is subordinate tog(z), written symbolically as follows:

f ≺g or f(z)≺g(z),

if there exists a Schwarz functionw(z), which (by definition) is analytic inU with w(0) = 0 and |w(z)| < 1, (z ∈ U), such that f(z) = g(w(z)) for all z ∈ U. In particular, if the function g(z) is univalent in U, then we have the following equivalence (cf., e.g., [10]; see also [11, p.4]):

f(z)≺g (z)⇔f(0)≺g(0) and f(U)⊂g(U).

2010 AMS Subject Classification: 30C45.

Keywords and phrases: Multiplier transformations; differential subordination;

superordination.

157

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Supposing thatpandhare two analytic functions inU, let ϕ(r, s, t;z) :C³×U →C.

Ifpandϕ(p(z), zp⁰(z), z²p⁰⁰(z);z) are univalent functions inU and ifpsatisfies the second-order superordination

h(z)≺ϕ(p(z), zp⁰(z), z²p⁰⁰(z);z), (1.3) then p is said to be a solution of the differential superordination (1.3). (If f is subordinate toF, thenF is superordinate tof). An analytic function qis called a subordinant of (1.3), if q(z) ≺ p(z) for all the functions p satisfying (1.3). A univalent subordinantqethat satisfiesq≺qefor all of the subordinantsqof (1.3), is called the best subordinant (cf., e.g.,[10], see also [11]).

Recently, Miller and Mocanu [12] obtained sufficient conditions on the func- tionsh,q andϕfor which the following implication holds:

h(z)≺ϕ(p(z), zp⁰(z), z²p⁰⁰(z);z)⇒q(z)≺p(z). (1.4) Using these results, Bulboaca [5] considered certain classes of first-order differential superordinations as well as superordination preserving integral operators [4]. Ali et al. [1], have used the results of Bulboaca [5] and obtained sufficient conditions for certain normalized analytic functionsf(z) to satisfy

q1(z)≺ zf⁰(z)

f(z) ≺q2(z), (1.5)

where q1 and q2 are given univalent functions in U with q1(0) = 1, Shanmugam et al. [17] obtained sufficient conditions for normalized analytic functionsf(z) to satisfy

q1(z)≺ f(z)

zf⁰(z) ≺q2(z), and q1(z)≺ z²f⁰(z)

{f(z)}² ≺q2(z),

whereq1 andq2 are given univalent functions inU withq1(0) = 1. and q2(0) = 1.

Liu [9] introduced and studied the class of functions B(β, α, ρ) defined by f ∈ B(β, α, ρ) if and only if

Re

½

(1−β)(f(z)

z )^α+βzf⁰(z) f(z) (f(z)

z )^α

¾

> ρ, wheref(z)∈A, β≥0, α >0 andρ≥0.

Many essentially equivalent definitions of multiplier transformation have been given in literature (see [7], [8], and [20]). In [6] Catas defined the operatorI^m(λ, `) as follows:

Definition 1 [6]. Let the function f(z) ∈ A(n). For m ∈ N0 = N∪ {0}, λ≥0, `≥ 0, the extended multiplier transformationI^m(λ, `) on A(n) is defined by the following infinite series:

I^m(λ, `)f(z) =z+ P^∞

k=n+1

·`+ 1 +λ(k−1)

`+ 1

¸_m

akz^k, m∈N0, z∈U. (1.6)

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We can write (1.6) as follows:

I^m(λ, `)f(z) = (Φ^,m_λ,`∗f)(z), where

Φ^m_λ,`(z) =z+ P^∞

k=n+1

·`+ 1 +λ(k−1)

`+ 1

¸_m z^k.

It is easily verified from (1.6), that

λz(I^m(λ, `)f(z))⁰= (1 +`)I^m+1(λ, `)f(z)−[1−λ+`]I^m(λ, `)f(z) (λ >0). (1.7) We note that:

I⁰(λ, `)f(z) =f(z) and I¹(1,0)f(z) =zf⁰(z).

Also by specializing the parametersλ, `andmwe obtain the following operators studied by various authors:

(i)I^m(1, `) =I^m(`)f(z) (see Cho and Srivastava [8] and Cho and Kim [7]);

(ii)I^m(λ,0)f(z) =D^m_λf(z) (see Al-Oboudi [2]);

(iii)I^m(1,0) =D^mf(z) (see Salagean [16]);

(iv)I^m(1,1) =I^mf(z) (see Uralegaddi and Somanatha [20]);

2. Preliminaries

In order to prove our subordination and superordination results, we make use of the following known definition and lemmas.

Definition 2. [12] Denote byQthe set of all functionsf(z) that are analytic and injective onU\E(f) where

E(f) ={ζ:ζ∈∂U and lim

z→ζf(z) =∞}, (2.1)

and are such thatf⁰(ζ)6= 0 forζ∈∂U\E(f).

Lemma 1. [11]Let the functionq(z)be univalent in the unit discU, and letθ andϕbe analytic in a domainD containingq(U), with ϕ(w)6= 0 whenw∈q(U).

Set Q(z) =zq⁰(z)ϕ(q(z)),h(z) =θ(q(z)) +Q(z)and suppose that (i)Qis a starlike function in U,

(ii) Re

µzh⁰(z) Q(z)

¶

>0 forz∈U.

If pis analytic inU withp(0) =q(0),p(U)⊆D and

θ(p(z)) +zp⁰(z)ϕ(p(z))≺θ(q(z)) +zq⁰(z)ϕ(q(z)), (2.2) thenp(z)≺q(z), andq is the best dominant.

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Lemma 2. [17]Let q be a convex function in U and let ψ∈Cwith δ∈C^∗ = C\{0} with

Re µ

1 + zq⁰⁰(z) q⁰(z)

¶

>max

½

0;−Reψ δ

¾

, z∈U.

If p(z)is analytic inU, and

ψp(z) +δzp⁰(z)≺ψq(z) +δzq⁰(z), (2.3) thenp(z)≺q(z), andq is the best dominant.

Lemma 3. [4]Let q(z)be a convex univalent function in the unit disc U and letθ andϕbe analytic in a domain D containingq(U). Suppose that

(i)Re

½θ⁰(q(z)) ϕ(q(z))

¾

>0 forz∈U; (ii) zq⁰(z)ϕ(q(z))is starlike inU.

Ifp∈H[q(0),1]∩Qwithp(U)⊆D, andθ(p(z)) +zp⁰(z)ϕ(p(z))is univalent inU, and

θ(q(z)) +zq⁰(z)ϕ(q(z))≺θ(p(z)) +zp⁰(z)ϕ(p(z)), thenq(z)≺p(z), andq is the best subordinant.

Lemma 4. [12]Let q be convex univalent inU and letδ∈C, with Re(δ)>0.

If p∈H[q(0),1]∩Q andp(z) +δzp⁰(z)is univalent inU, then

q(z) +δzq⁰(z)≺p(z) +δzp⁰(z), (2.4) impliesq(z)≺p(z) (z∈U), and qis the best subordinant.

This last lemma gives us a necessary and sufficient condition for the univalence of a special function which will be used in some particular cases:

Lemma 5. [15]The function q(z) = (1−z)^−2ab is univalent in U if and only if |2ab−1| ≤1 or|2ab+ 1| ≤1.

3. Subordination results for analytic functions

Unless otherwise mentioned we shall assume throughout the paper thatβ∈C^∗, α >0,λ >0,`≥0,n∈N,m∈N0 and the powers understood as principle values.

Theorem 1. Let q(z)be convex univalent in U, withq(0) = 1. Suppose that Re

µ

1 +zq⁰⁰(z) q⁰(z)

¶

>max

½

0;−Reα β

¾

. (3.1)

If f(z)∈A(n)satisfies the subordination:

Φ(f, m, λ, `, β, α)≺q(z) +β

αzq⁰(z), (3.2)

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where

Φ(f, m, λ, `, β, α) =

·

1−β(`+ 1 λ )

¸ µI^m(λ, `)f(z) z

¶_α

+β µ`+ 1

λ

¶I^m+1(λ, `)f(z) I^m(λ, `)f(z)

µI^m(λ, `)f(z) z

¶_α

, (3.3)

then µ

I^m(λ, `)f(z) z

¶_α

≺q(z), (3.4)

andq(z)is the best dominant of(3.2).

Proof. Define the functionp(z) by p(z) =

µI^m(λ, `)f(z) z

¶_α

(z∈U). (3.5)

Thenp(z) is analytic inU andp(0) = 1. Differentiating (3.5) logarithmically with respect toz, and using the identity (1.7) in the resulting equation, we have

· 1−β

µ`+ 1 λ

¶¸ µI^m(λ, `)f(z) z

¶_α +β

µ`+ 1 λ

¶I^m+1(λ, `)f(z) I^m(λ, `)f(z) ×

×

µI^m(λ, `)f(z) z

¶_α

=p(z) +β

αzp⁰(z). (3.6) Thus the subordination (3.2) is equivalent to

p(z) +β

αzp⁰(z)≺q(z) +β

αzq⁰(z). (3.7)

Applying Lemma 2 withγ= β

α (α >0), the proof of Theorem 1 is completed.

Remark 1. Putting m = ` = 0, λ = n = 1 and β ≥ 0 in Theorem 1, we obtain the result obtained by Shanmungam et al. [18, Theorem 3.1].

Puttingλ= 1 and`= 0 in Theorem 1, we obtain the following corollary.

Corollary 1. Let q(z) be convex univalent inU, withq(0) = 1and suppose that q(z)satisfies the condition (3.1). Iff(z)∈A(n)satisfies the subordination:

Φ(f, m, β, α)≺q(z) +β αzq⁰(z), where

Φ(f, m, β, α) = (1−β)

µD^mf(z) z

¶_α

+βD^m+1f(z) D^mf(z)

µD^mf(z) z

¶_α

, (3.8)

then

µD^mf(z) z

¶_α

≺q(z)andq(z)is the best dominant.

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Putting`= 0 in Theorem 1, we obtain the following corollary.

Corollary 2. Let q(z) be convex univalent inU, withq(0) = 1and suppose that q(z)satisfy the condition (3.1). Iff(z)∈A(n)satisfies the subordination

Φ(f, m, λ, β, α)≺q(z) +β

αzq⁰(z), where

Φ(f, m, λ, β, α) = µ

1−β λ

¶ µD^m_λf(z) z

¶_α +β

λ

D_λ^m+1f(z) D^m_λf(z)

µD^m_λf(z) z

¶_α

, (3.9)

then

µD_λ^mf(z) z

¶_α

Puttingλ= 1 in Theorem 1, we obtain the following corollary.

Corollary 3. Let q(z) be convex univalent inU, withq(0) = 1and suppose that q(z)satisfy (3.1). Iff(z)∈A(n)satisfies the subordination

Φ(f, m, `, β, α)≺q(z) +β

αzq⁰(z), where

Φ(f, m, `, β, α) = [1−β(`+ 1)]

µI^m(`)f(z) z

¶_α + +β(`+ 1)I^m+1(`)f(z)

I^m(`)f(z)

µI^m(`)f(z) z

¶_α

, (3.10)

then

µI^m(`)f(z) z

¶_α

Takingq(z) =_1+Bz^1+Az (−1≤B < A≤1) in Theorem 1, we obtain the following corollary.

Corollary 4. Let −1≤B < A≤1 and suppose that Re

½1−Bz 1 +Bz

¾

>max

½

0,−Reα β

¾ .

If f(z)∈A(n)satisfies the subordination Φ(f, m, λ, `, β, α)≺ 1 +Az

1 +Bz +β α

(A−B)z (1 +Bz)² , whereΦ(f, m, λ, `, β, α)is given by (3.3), then

µI^m(λ, `)f(z) z

¶_α

≺ 1 +Az 1 +Bz and 1 +Az

1 +Bz is the best dominant.

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Remark 2. Putting m = ` = 0, λ = n = 1 and β ≥ 0 in Corollary 4, we obtain the result obtained by Shanmungam et al. [18, Corollary 3.2].

Theorem 2. Let q(z) be univalent in U, and α, γ ∈ C. Suppose that q(z) satisfies

Re

½

1 + zq⁰⁰(z)

q⁰(z) −zq⁰(z) q(z)

¾

>0. (3.11)

If f(z)∈A(n)satisfies the subordination

ψ(f, m, λ, `, β, α)≺1 +γzq⁰(z)

q(z) , (3.12)

where

ψ(f, m, λ, `, β, α) = 1 +γα µ`+ 1

λ

¶ ·I^m+1(λ, `)f(z) I^m(λ, `)f(z) −1

¸

, (3.13) then

µI^m(λ, `)f(z) z

¶_α

≺q(z)and q(z)is the best dominant.

Proof. Letp(z) be defined by (3.5). Then, simple computations show that zp⁰(z)

p(z) =α µ`+ 1

λ

¶ ·I^m+1(λ, `)f(z) I^m(λ, `)f(z) −1

¸ . Puttingθ(w) = 1 andϕ(w) = γ

w, we can observe thatθ(w) is analytic inC, ϕ(w) is analytic inC^∗ andϕ(w)6= 0 (w∈C^∗). If

ψ(z) =zq⁰(z) =ϕ(q(z)) =γzq⁰(z) q(z) and

h(z) =θ(q(z)) +ψ(z) = 1 +γzq⁰(z) q(z) , then, from (3.11), we find thatψ(z) is starlike univalent inU and

Re

µzh⁰(z) ψ(z)

¶

= Re

½

1 + zq⁰⁰(z)

q⁰(z) −zq⁰(z) q(z)

¾

>0.

Then applying Lemma 1, the proof is completed.

Remark 3. Takingm=` = 0 andλ=n= 1 in Theorem 2, we obtain the result obtained by Shanmugam et al. [18, Theorem 3.4].

Corollary 5. Assume that (3.11)holds. Iff(z)∈A(n), and 1 +γα

·D^m+1f(z) D^mf(z) −1

¸

≺1 +γzq⁰(z) q(z) , then

µD^mf(z) z

¶_α

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Corollary 6. Assume that (3.11)holds. Iff(z)∈A(n), and 1 + γα

λ

·D_λ^m+1f(z) D^m_λf(z) −1

¸

≺1 +γzq⁰(z) q(z) , then

µD_λ^mf(z) z

¶_α

Corollary 7. Assume that (3.11)holds. Iff(z)∈A(n), and 1 +γα(`+ 1)

·I^m+1(`)f(z) I^m(`)f(z) −1

¸

≺1 +γzq⁰(z) q(z) , then

µI^m(`)f(z) z

¶α

Taking q(z) = _(1−z)¹2αb (α, b ∈ C^∗), γ = _αb¹, λ = n = 1 and m = ` = 0 in Theorem 2, we obtain the next result due to Obradovi´c et al. [13, Theorem 1].

Corollary 8. [13]Letα, b∈C^∗ such that|2αb−1| ≤1 or|2αb+ 1| ≤1. Let f(z)∈A and suppose that ^f(z)_z 6= 0 for allz∈U. If

1 +1 b

µzf⁰(z) f(z) −1

¶

≺1 +z 1−z, then

µf(z) z

¶_α

≺(1−z)^−2αb and(1−z)^−2αb is the best dominant.

Remark 4. Forα= 1, Corollary 8 reduces to the recent result of Srivastava and Lashin [19, Corollary 1].

Taking q(z) = (1 +Bz)^α(A−B)^B , −1 ≤ B < A ≤1, B 6= 0, α ∈ C^∗, γ = 1, m=`= 0 and λ= 1 in Theorem 2, we obtain the following corollary.

Corollary 9. Let −1≤B < A≤1, withB6= 0, and suppose that

¯¯

¯¯α(A−B)

B −1

¯¯

¯¯≤1 or

¯¯

¯¯α(A−B)

B + 1

¯¯

¯¯≤1.

If f(z)∈A(n)such that ^f(z)_z 6= 0 for allz∈U, and letα∈C^∗. If 1 +α

¶

≺ 1 + [B+α(A−B)]z

1 +Bz ,

then µ

f(z) z

¶_α

≺(1 +Bz)^α(A−B)^B , and(1 +Bz)^α(A−B)^B is the best dominant.

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Remark 5. For α = n = 1, Corollary 9 reduces to the recent result of Obradovi´c and Owa [14].

Puttingq(z) = (1−z)^−2αb^cos^λe^−iλ (α, b∈C^∗;|λ|<^π₂),γ= _αb^e_cos^iλ _λ,n=λ= 1 and m = ` = 0 in Theorem 2, we obtain the next result due to Aouf et al. [3, Theorem 1].

1 + e^iλ bcosλ

¶

≺ 1 +z 1−z ,

then µ

f(z) z

¶_α

≺(1−z)^−2αb^cos^λe^−iλ, and(1−z)^−2αb^cos^λe^−iλ is the best dominant.

4. Superordination and Sandwich results

Theorem 3. Let q(z) be convex in U with q(0) = 1, and β ∈C, Reβ > 0.

If f(z) ∈ A(n) such that (^I^m^(λ,`)f(z)_f(z) )^α ∈ H[q(0),1]∩Q and Φ(f, m, λ, `, β, α) is univalent inU and satisfies the superordination:

q(z) +β

αzq⁰(z)≺Φ(f, m, λ, `, β, α), (4.1) whereΦ(f, m, λ, `, β, α)is given by (3.3), then

q(z)≺

µI^m(λ, `)f(z) z

¶_α

andq(z)is the best subordinant.

Proof. Letp(z) be given by (3.5) and proceeding as in the proof of Theorem 1, the subordination (4.1) becomes

q(z) +β

αzq⁰(z)≺p(z) +β αzp⁰(z).

The proof follows by an application of Lemma 4.

Theorem 4. Let q(z) be convex univalent in U, β ∈ C, Re(β) > 0, and (^I^m^(λ,`)f(z)_z )^α∈H[q(0),1]∩Q. Iff(z)∈A(n)and

1 +γzq⁰(z)

q(z) ≺1 +γα µ`+ 1

λ

¶ ·I^m+1(λ, `)f(z) I^m(λ, `)f(z) −1

¸

, (4.2)

then

q(z)≺

µI^m(λ, `)f(z) z

¶_α

andq(z)is the best subordinant.

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Remark 6. Puttingm=`= 0,λ=n= 1 in Theorem 4, we obtain the result obtained by Shanmugam et al. [18, Theorem 4.3].

Combining Theorem 1 with Theorem 3 and Theorem 2 with Theorem 4, we state the following “Sandwich results”.

Theorem 5. Letq1, q2be convex inU withq1(0) =q2(0) = 1,β ∈C,Reβ >0 and satisfies(3.1). Iff(z)∈A(n),(^I^m^(λ,`)f(z)_z )^α∈H[q(0),1]∩Q,Φ(f, m, λ, `, β, α) is univalent in the unit discU, whereΦ(f, m, λ, `, β, α)is defined by(3.3) and

q1(z) +β

αzq₁⁰(z)≺Φ(f, m, λ, `, β, α)≺q2(z) +β

αzq₂⁰(z), (4.3) then

q1(z)≺

µI^m(λ, `)f(z) z

¶_α

≺q2(z)

andq1(z)andq2(z)are, respectively, the best subordinant and best dominant.

Corollary 11. Let q1(z), q2(z) be convex in U with q1(0) = q2(0) = 1, β ∈C,Reβ > 0 and satisfies (3.1). If f(z)∈ A(n),(^D^m_z^f(z))^α ∈H[q(0),1]∩Q, Φ(f, m, β, α)is univalent in the unit discU, whereΦ(f, m, β, α)is defined by(3.8) and

q1(z) +β

αzq₁⁰(z)≺Φ(f, m, β, α)≺q2(z) + β αzq₂⁰(z), then

q1(z)≺

µD^mf(z) z

¶_α

≺q2(z)

Corollary 12. Let q1(z), q2(z) be convex in U with q1(0) = q2(0) = 1, β ∈C,Reβ > 0 and satisfies (3.1). If f(z)∈ A(n),(^D^m^λ_z^f(z))^α ∈H[q(0),1]∩Q, Φ(f, m, λ, β, α)is univalent in the unit discU, whereΦ(f, m, λ, β, α) is defined by (3.9) and

q1(z) +β

αzq⁰₁(z)≺Φ(f, m, λ, β, α)≺q2(z) +β αzq⁰₂(z), then

q1(z)≺

µD^m_λf(z) z

¶_α

≺q2(z)

Corollary 13. Let q1(z), q2(z) be convex in U with q1(0) = q2(0) = 1, β ∈C,Reβ >0 and satisfies (3.1). If f(z)∈A(n),(^I^m^(`)f(z)_z )^α∈H[q(0),1]∩Q,

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Φ(f, m, `, β, α) is univalent in the unit disc U, where Φ(f, m, `, β, α) is defined by (3.10) and

q1(z) +β

αzq₁⁰(z)≺Φ(f, m, `, β, α)≺q2(z) +β αzq₂⁰(z), then

q₁(z)≺

µI^m(`)f(z) z

¶_α

≺q₂(z)

Theorem 6. Let q1(z), q2(z) be convex inU with q1(0) =q2(0) = 1,β ∈C, Reβ > 0 and satisfies (3.1). If f(z) ∈ A(n), (^I^m^(λ,`)f(z)_z )^α ∈ H[q(0),1]∩Q, ψ(f, m, λ, `, β, α)is univalent in the unit discU, where ψ(f, m, λ, `, β, α)is defined by(3.13)and

1 +γzq⁰₁(z)

q1(z) ≺ψ(f, m, λ, `, β, α)≺1 +γzq⁰₂(z)

q2(z) , (4.4) then

q1(z)≺

µI^m(λ, `)f(z) z

¶_α

≺q2(z)

andq1(z)andq2(z)are, respectively, the best subordinant and the best dominant.

Corollary 14. Letq₁(z), q₂(z)be convex inU withq₁(0) =q₂(0) = 1,β ∈C, Reβ >0 and satisfies(3.1). Iff(z)∈A(n),(^D^m_z^f(z))^α∈H[q(0),1]∩Q,

1 +γα

·D^m+1f(z) D^mf(z) −1

¸

is univalent inU and 1 +γzq₁⁰(z)

q1(z) ≺1 +γα

·D^m+1f(z) D^mf(z) −1

¸

≺1 +γzq₂⁰(z) q2(z), then

q1(z)≺

µD^mf(z) z

¶_α

≺q2(z)

Corollary 15. Letq1(z), q2(z)be convex inU withq1(0) =q2(0) = 1,β ∈C, Reβ >0 and satisfies(3.1). Iff(z)∈A(n),(^D^m^λ_z^f(z))^α∈H[q(0),1]∩Q,

1 + γα λ

·D_λ^m+1f(z) D^m_λf(z) −1

¸

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is univalent inU and 1 +γzq⁰₁(z)

q1(z) ≺1 + γα λ

·D_λ^m+1f(z) D^m_λf(z) −1

¸

≺1 +γzq₂⁰(z) q2(z) , then

q1(z)≺

µD^m_λf(z) z

¶_α

≺q2(z)

Corollary 16. Letq1(z), q2(z)be convex inU withq1(0) =q2(0) = 1,β ∈C, Reβ >0 and satisfies(3.1). Iff(z)∈A(n),(^I^m^(`)f(z)_z )^α∈H[q(0),1]∩Q,

1 +γα(`+ 1)

·I^m+1(`)f(z) I^m(`)f(z) −1

¸

is univalent inU and 1 +γzq₁⁰(z)

q1(z) ≺1 +γα(`+ 1)

·I^m+1(`)f(z) I^m(`)f(z) −1

¸

≺1 +γzq⁰₂(z) q2(z), then

q1(z)≺

µI^m(`)f(z) z

¶_α

≺q2(z)

Remark 7. Puttingm=`= 0,λ=n= 1 andβ ≥0 in Theorem 6, we obtain the following result which improves the result of Shanmugam et al. [18, Theorem 5.2].

Corollary 17. Let q1(z), q2(z) be convex in U with q1(0) = q2(0) = 1, β ∈ C, Reβ > 0 and satisfies (3.1). If f(z) ∈ A(n), (^f(z)_z )^α ∈ H[q(0),1]∩Q, 1 +γα(^zf_f(z)⁰^(z)−1) is univalent inU and

1 +γzq₁⁰(z)

q1(z) ≺1 +γα

¶

≺1 +γzq⁰₂(z) q2(z), then

q₁(z)≺ µf(z)

z

¶_α

≺q₂(z)

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(received 11.03.2010; in revised form 04.02.2011)

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt E-mail:[email protected], [email protected], r [email protected],

ekram [email protected]