DIFFERENTIAL SANDWICH THEOREMS FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH SRIVASTAVA-ATTIYA OPERATOR

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FOR CERTAIN SUBCLASSES

OF ANALYTIC FUNCTIONS ASSOCIATED WITH SRIVASTAVA-ATTIYA OPERATOR

G. MURUGUSUNDARAMOORTHY, N. MAGESH and K. THILAGAVATHI

The main purpose of the present paper is to investigate some subordination- preserving and superordination results involving Hadamard product for certain analytic functions preserving properties of a certain family of integral operators.

Relevant connections of the results are remarked and the new results are also pointed out.

AMS 2010 Subject Classification: Primary 30C45; Secondary 30C80.

Key words: univalent functions, starlike functions, convex functions, differential subordination, differential superordination, Hadamard product (convolution), Hurwitz-Lerch Zeta function, Srivastava-Attiya operator, Choi-Saigo-Srivastava operator.

1. INTRODUCTION

LetHbe the class of analytic functions inU:={z:|z|<1}and H[a, n]

be the subclass of Hconsisting of functions of the form f(z) =a+anzⁿ+an+1zⁿ⁺¹+· · · .

LetAbe the subclass of Hconsisting of functions of the form (1.1) f(z) =z+a₂z²+· · ·=z+

∞

X

n=2

a_nzⁿ.

For two functions f(z) ∈ A and g(z) = z +

∞

P

n=2

bnzⁿ, the Hadamard product (or convolution) of f and g is defined by

(1.2) (f∗g)(z) :=z+

∞

X

n=2

a_nb_nzⁿ=: (g∗f)(z).

REV. ROUMAINE MATH. PURES APPL.,57(2012),3, 257-273

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We recall here a general Hurwitz-Lerch Zeta function Φ(z, s, a) defined in [32] by

(1.3) Φ(z, s, a) :=

∞

X

n=0

zⁿ (n+a)^s

(a∈C\ {Z⁻₀}; s∈C, when |z|<1; R(s)>1 and|z|= 1),

where, as usual, Z⁻₀ :=Z\ {N},(Z:={0,±1,±2,±3, . . .}); N:={1,2,3, . . .}.

Several interesting properties and characteristics of the Hurwitz-Lerch Zeta function Φ(z, s, a) can be found in the recent investigations by Choi and Srivastava [10], Ferreira and Lopez [12], Garg et al. [14], Lin and Srivastava [16], Lin et al. [17], and others. Srivastava and Attiya [31] introduced and investigated the linear operator: J_s,b:A → Adefined in terms of the Hadamard product by

(1.4) J_s,bf(z) =G_s,b∗f(z)

(z∈U; b∈C\ {Z⁻₀}; s∈C; f ∈ A), where, for convenience, (1.5) G_s,b(z) := (1 +b)^s[Φ(z, s, b)−b^−s] (z∈U).

We recall here the following relationships which follow easily by using (1.4) and (1.5)

(1.6) J_b^sf(z) =z+

∞

X

n=2

1 +b n+b

s

a_nzⁿ.

Motivated essentially by the Srivastava-Attiya operator, Al-Shaqsi and Darus [1] introduced and investigated the integral operator

(1.7) J_s,b^λ,µf(z) =z+

∞

X

n=2

1 +b n+b

s

λ!(n+µ−2)!

(µ−2)!(n+λ−1)!a_nzⁿ (z∈U), where (and throughout this paper unless otherwise mentioned) the parameters µ, b are constrained as b∈C\ {Z⁻₀};s∈C,µ≥0 andλ >−1. Further note that J_s,b^1,2 is the Srivastava-Attiya operator, and J_0,b^λ,µ is the well-known Choi- Saigo-Srivastava operator [9].

Assumingλ= 1 andµ= 2, we state the following integral operators by specializing s andb.

(1) Fors= 0

(1.8) J_b⁰(f)(z) :=f(z).

(2) Fors= 1 andb= 0

(1.9) J₀¹(f)(z) :=

Z z 0

f(t)

t dt:=Lf(z).

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(3) Fors= 1 andb=ν (ν >−1) (1.10) J_ν¹(f)(z) :=F_ν(f)(z) = 1 +ν

z^ν Z z

0

t^ν−1f(t)dt:=z+

∞

X

n=2

1 +ν n+ν

anzⁿ.

(4) Fors=σ (σ >0) and b= 1 (1.11) J₁^σ(f)(z) :=z+

∞

X

n=2

2 n+ 1

σ

a_nzⁿ=I^σ(f)(z),

where L(f) and F_ν are the familiar Alexander and Bernardi integral operators respectively, and I^σ(f) is the Jung-Kim-Srivastava integral operator [15]

closely related to some multiplier transformation studied by Flett [13]. It is easy to verify from (1.7),

z

J_{s, b}^{λ+1, µ}f(z) 0

(z) = (λ+ 1)J_s,b^λ,µf(z)−λJ_{s, b}^{λ+1, µ}f(z), (1.12)

z

J_{s+1, b}^{λ, µ} f(z)0

(z) = (b+ 1)J_s,b^λ,µf(z)−bJ_{s+1, b}^{λ, µ} f(z), (1.13)

z

J_s,b^λ,µf(z)0

(z) =µJ_s,b^λ,µ+1f(z)−(µ−1)J_s,b^λ,µf(z).

(1.14)

Let p, h ∈ H and let φ(r, s, t;z) : C³ ×U → C. If p and φ p(z), zp⁰(z), z²p⁰⁰(z);z

are univalent and ifp satisfies the second order superordination (1.15) h(z)≺φ(p(z), zp⁰(z), z²p⁰⁰(z);z),

then p is a solution of the differential superordination (1.15). (If f is subor- dinate to F, thenF is superordinate to f.) An analytic functionq is called a subordinant ifq≺pfor allpsatisfying (1.15). A univalent subordinantqethat satisfies q ≺qefor all subordinants q of (1.15) is said to be the best subordinant. Recently, Miller and Mocanu [21] obtained conditions on h,q and φfor which the following implication holds

h(z)≺φ(p(z), zp⁰(z), z²p⁰⁰(z);z)⇒q(z)≺p(z).

In [5] Bulboac˘a (see also [4]) considered certain classes of first order differential superordinations as well as superordination-preserving integral operators by using the results of Miller and Mocanu [21]. Further, using the results in [4] and [21], Aouf and Mostafa [2], Magesh et al. [18, 19], Mostafa and Aouf [22], Murugusundaramoorthy and Magesh [23, 24, 25], and Selvaraj and Karthikeyan [30] have obtained sandwich results for certain classes of analytic functions.

Making use of the operatorJ_s,b^λ,µ,in this paper we find sufficient condition for certain normalized analytic functions f(z) in U such that (f ∗Ψ)(z) 6= 0

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and f to satisfy

(1.16) q1(z)≺ αJ_s,b^λ,µ+1(f ∗Φ)(z) +βJ_s,b^λ,µ(f ∗Ψ)(z) (α+β)z

!η

≺q2(z), where q₁, q₂ are given univalent functions inU withq₁(0) = 1, q₂(0) = 1 and Φ(z) = z+

∞

P

n=2

λ_nzⁿ, Ψ(z) = z+

∞

P

n=2

µ_nzⁿ are analytic functions in U with λ_n≥0, µ_n≥0 andλ_n≥µ_n.Also, we obtain the number of known results as their special cases.

2. BASIC DEFINITIONS AND LEMMAS For our present investigation, we shall need the following:

Lemma2.1 ([27, p. 159, Theorem 6.2]). The functionL(z, t) =a₁(t)z+ a2(t)z²+· · · witha1(t)6= 0fort≥0and lim

t→∞|a₁(t)|= +∞,is a subordination chain if

Re (

z

∂L(z,t)

∂z

∂L(z,t)

∂t

)

>0, z∈U, t≥0.

Definition 2.1 ([21, p. 817, Definition 2]). Denote by Q, the set of all functions f that are analytic and injective on U−E(f), where

E(f) =n

ζ ∈∂U: lim

z→ζf(z) =∞o and are such that f⁰(ζ)6= 0 for ζ∈∂U−E(f).

Lemma 2.2 ([20, p. 132, Theorem 3.4h]). Let q be univalent in the unit disk Uand θ andφ be analytic in a domain Dcontaining q(U) with φ(w)6= 0 when w∈q(U). Set

Q(z) :=zq⁰(z)φ(q(z)) and h(z) :=θ(q(z)) +Q(z).

Suppose that

(1)Q(z) is starlike univalent in Uand (2) Ren_zh0(z)

Q(z)

o

>0 for z∈U.

If p is analytic with p(0) =q(0), p(U)⊆D and

(2.1) θ(p(z)) +zp⁰(z)φ(p(z))≺θ(q(z)) +zq⁰(z)φ(q(z)), then

p(z)≺q(z) and q is the best dominant.

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Lemma 2.3 ([5, p. 289, Corollary 3.2]). 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) Re{ϑ⁰(q(z))/ϕ(q(z))}>0 for z∈U and (2)ψ(z) =zq⁰(z)ϕ(q(z))is starlike univalent in U.

If p(z) ∈H[q(0),1]∩Q, with p(U) ⊆ D, and ϑ(p(z)) +zp⁰(z)ϕ(p(z)) is univalent in Uand

(2.2) ϑ(q(z)) +zq⁰(z)ϕ(q(z))≺ϑ(p(z)) +zp⁰(z)ϕ(p(z)), then q(z)≺p(z) and q is the best subordinant.

3. SUBORDINATION RESULTS Using Lemma 2.2, we first prove the following theorem.

Theorem 3.1. Let Φ,Ψ∈A, γi ∈C (i= 1, . . . ,4) (γ4 6= 0), η, α, β ∈ C such that η6= 0 and α+β 6= 0,and q be convex univalent with q(0) = 1,and assume that

(3.1) Re γ₃

γ₄q(z) +2γ₂

γ₄ q²(z)−zq⁰(z)

q(z) + 1 + zq⁰⁰(z) q⁰(z)

>0 (z∈U).

If f ∈A satisfies

∆^(γⁱ⁾⁴¹(f; Φ,Ψ) = ∆(f, Φ,Ψ, γ1, γ2, γ3, γ4)≺γ1+γ2q²(z) +γ3q(z) +γ4

zq⁰(z) q(z) , (3.2)

where

∆^(γⁱ⁾⁴¹(f; Φ,Ψ) :=

(3.3)











γ1+γ2

αJ_s,b^λ,µ+1(f∗Φ)(z)+βJ_s,b^λ,µ(f∗Ψ)(z) (α+β)z

2η

+γ3

η

+γ₄η

α(µ+1)[J_s,b^λ,µ+2(f∗Φ)(z)−J_s,b^λ,µ+1(f∗Φ)(z)]+βµ[J_s,b^λ,µ+1(f∗Ψ)(z)−J_s,b^λ,µ(f∗Ψ)(z)]

αJ_s,b^λ,µ+1(f∗Φ)(z)+βJ_s,b^λ,µ(f∗Ψ)(z)

,

then

αJ_s,b^λ,µ+1(f∗Φ)(z) +βJ_s,b^λ,µ(f ∗Ψ)(z) (α+β)z

!η

≺q(z) and q is the best dominant.

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Proof. Define the functionp by

(3.4) p(z) := αJ_s,b^λ,µ+1(f∗Φ)(z) +βJ_s,b^λ,µ(f ∗Ψ)(z) (α+β)z

!η

(z∈U).

Then, the function p is analytic inU andp(0) = 1. Therefore, by making use of (3.4) and (1.12), we obtain

γ₁+γ₂p²(z) +γ₃p(z) +γ₄zp⁰(z) p(z) = (3.5)









 γ1+γ2

2η

+γ3

η

+γ₄η

α(µ+1)[J_s,b^λ,µ+2(f∗Φ)(z)−J_s,b^λ,µ+1(f∗Φ)(z)]+βµ[J_s,b^λ,µ+1(f∗Ψ)(z)−J_s,b^λ,µ(f∗Ψ)(z)]

αJ_s,b^λ,µ+1(f∗Φ)(z)+βJ_s,b^λ,µ(f∗Ψ)(z)

.

By using (3.5) in (3.2), we have

(3.6) γ₁+γ₂p²(z) +γ₃p(z) +γ₄zp⁰(z)

p(z) ≺γ₁+γ₂q²(z) +γ₃q(z) +γ₄zq⁰(z) q(z) . By setting

θ(w) :=γ1+γ2ω²(z) +γ3ω and φ(ω) := γ4

w,

it can be easily observed thatθ(w), φ(w) are analytic inC−{0}andφ(w)6= 0.

Also, we see that

Q(z) :=zq⁰(z)φ(q(z)) =γ₄zq⁰(z) q(z) and

h(z) :=θ(q(z)) +Q(z) =γ1+γ2q²(z) +γ3q(z) +γ4

zq⁰(z) q(z) . It is clear that Q(z) is starlike univalent in Uand

Re

zh⁰(z) Q(z)

= Re γ₃

γ4

q(z) +2γ₂ γ4

q²(z)−zq⁰(z)

q(z) + 1 +zq⁰⁰(z) q⁰(z)

>0.

By the hypothesis of Theorem 3.1, the result now follows by an application of Lemma 2.2.

By fixing Φ(z) = _1−z^z and Ψ(z) = _1−z^z in Theorem 3.1, we obtain the following corollary.

Corollary3.1. Let γ_i ∈C(i= 1, . . . ,4) (γ₄ 6= 0), η, α, β∈ C such that η 6= 0 andα+β 6= 0,and q be convex univalent with q(0) = 1,and(3.1)holds

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true. If f ∈Asatisfies

γ₁+γ₂ αJ_s,b^λ,µ+1f(z) +βJ_s,b^λ,µf(z) (α+β)z

!2η

+γ₃ αJ_s,b^λ,µ+1f(z) +βJ_s,b^λ,µf(z) (α+β)z

!η

+γ₄η α(µ+ 1)[J_s,b^λ,µ+2f(z)− J_s,b^λ,µ+1f(z)] +βµ[J_s,b^λ,µ+1f(z)− J_s,b^λ,µf(z)]

αJ_s,b^λ,µ+1f(z) +βJ_s,b^λ,µf(z)

!

≺γ₁+γ₂q²(z) +γ₃q(z) +γ₄zq⁰(z) q(z) , then

αJ_s,b^λ,µ+1f(z) +βJ_s,b^λ,µf(z) (α+β)z

!η

≺q(z)

and q is the best dominant.

By takings= 0 in Corollary 3.1, we state the following corollary.

Corollary3.2. Let γi ∈C(i= 1, . . . ,4) (γ4 6= 0), η, α, β∈ C such that η 6= 0 andα+β 6= 0,and q be convex univalent with q(0) = 1,and(3.1)holds true. If f ∈Asatisfies

γ1+γ2

αI_λ^µ+1f(z) +βI_λ^µf(z) (α+β)z

!2µ

+γ3

!η

+γ4η α(µ+ 1)[I_λ^µ+2f(z)− I_λ^µ+1f(z)] +βµ[I_λ^µ+1f(z)− I_λ^µf(z)]

αI_λ^µ+1f(z) +βI_λ^µf(z)

!

≺γ₁+γ₂q²(z) +γ₃q(z) +γ₄zq⁰(z) q(z) , then

!η

≺q(z),

where I_λ^µf(z) is the Choi-Saigo-Srivastava operator[9] andq is the best dominant.

By takings=α= 0, λ= 1, µ= 2 and β = 1 in Theorem 3.1, we state the following corollary.

Corollary 3.3. Let Ψ∈A, γ_i ∈C (i= 1, . . . ,4) (γ₄ 6= 0), η∈ C such that η 6= 0 and q be convex univalent with q(0) = 1, and (3.1) holds true. If

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f ∈A satisfies

γ1+γ2

(f ∗Ψ)(z) z

2η

+γ3

(f∗Ψ)(z) z

η

+γ4η

z(f∗Ψ)⁰(z) (f∗Ψ)(z) −1

≺γ₁+γ₂p²(z) +γ₃p(z) +γ₄zp⁰(z) p(z) ,

then

(f ∗Ψ)(z) z

η

≺q(z)

By fixing Ψ(z) = _1−z^z in Corollary 3.3, we obtain the following corollary.

Corollary 3.4. Let γi ∈ C (i = 1, . . . ,4) (γ4 6= 0), η ∈ C such that η 6= 0and q be convex univalent withq(0) = 1, and (3.1)holds true. If f ∈A satisfies

γ₁+γ₂ f(z)

z 2η

+γ₃ f(z)

z η

+γ₄η

zf⁰(z) f(z) −1

≺γ1+γ2p²(z) +γ3p(z) +γ4

zp⁰(z) p(z) ,

then

f(z) z

η

≺q(z)

By takingq(z) = _1+Bz^1+Az (−1≤B < A≤1) in Theorem 3.1, we have the following corollary.

Corollary3.5. LetΦ,Ψ∈A, γi ∈C(i= 1, . . . ,4) (γ4 6= 0), η, α, β∈ C such that η 6= 0 and α +β 6= 0, and q be convex univalent with q(0) = 1.

Assume that Re

(γ₃ γ4

1 +Az 1 +Bz

+2γ₂

γ4

1 +Az 1 +Bz

2

+ 1−ABz² (1 +Az)(1 +Bz)

)

>0.

If f ∈A and

∆^(γⁱ⁾⁴¹(f; Φ,Ψ)≺γ1+γ2

1 +Az 1 +Bz

2

+γ3

1 +Az 1 +Bz +γ4

(A−B)z (1 +Az)(1 +Bz), then

αJ_s,b^λ,µ+1(f∗Φ)(z) +βJ_s,b^λ,µ(f ∗Ψ)(z) (α+β)z

!η

≺ 1 +Az 1 +Bz and ^1+Az_1+Bz is the best dominant.

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By puttingγ1 =γ4 =β = 1, α=γ2 =γ3 = 0,Φ(z) = Ψ(z) = _1−z^z and q(z) = (1 +Bz)^η(A−B)/B in Theorem 3.1, we state the following corollary.

Corollary 3.6. If f ∈Asatisfies 1 +η

zf⁰(z) f(z) −1

≺ 1 +Az 1 +Bz, then

f(z) z

η

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

4. SUPERORDINATION RESULTS

Now, by applying Lemma 2.3, we prove the following theorem.

Theorem 4.1. Let Φ,Ψ∈A, γ_i ∈C (i= 1, . . . ,4) (γ₄ 6= 0), η, α, β ∈ C such that η6= 0 and α+β 6= 0,and q be convex univalent with q(0) = 1,and assume that

(4.1) Re

γ₃ γ4

q(z) +2γ₂ γ4

q²(z)

≥0.

If f ∈A,

η

∈H[q(0),1]∩Q.Let ∆^(γⁱ⁾⁴¹(f; Φ,Ψ) be univalent in Uand

(4.2) γ₁+γ₂q²(z) +γ₃q(z) +γ₄zq⁰(z)

q(z) ≺∆^(γⁱ⁾⁴¹(f; Φ,Ψ), where ∆^(γⁱ⁾⁴¹(f; Φ,Ψ) is given by(3.3), then

q(z)≺ αJ_s,b^λ,µ+1(f∗Φ)(z) +βJ_s,b^λ,µ(f ∗Ψ)(z) (α+β)z

!^η

and q is the best subordinant.

Proof. Define the functionp by

(4.3) p(z) := αJ_s,b^λ,µ+1(f∗Φ)(z) +βJ_s,b^λ,µ(f∗Ψ)(z) (α+β)z

!η

.

Simple computation from (4.3), we get,

∆^(γⁱ⁾⁴¹(f; Φ,Ψ) =γ₁+γ₂p²(z) +γ₃p(z) +γ₄zp⁰(z) p(z) ,

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then

γ₁+γ₂q²(z) +γ₃q(z) +γ₄zq⁰(z)

q(z) ≺γ₁+γ₂p²(z) +γ₃p(z) +γ₄zp⁰(z) p(z) . By settingϑ(w) =γ₁+γ₂w²+γ₃wand φ(w) = ^γ_w⁴,it is easily observed that ϑ(w) is analytic in C.Also,φ(w) is analytic in C− {0} and φ(w)6= 0.

If we let

L(z, t) =ϑ(q(z)) +φ(q(z))tzq⁰(z) =γ₁+γ₂q²(z) +γ₃q(z) +γ₄tzq⁰(z) q(z)

=a1(t)z+· · · . (4.4)

Differentiating (4.4) with respect to z andt, we have

∂L(z, t)

∂z = 2γ2q(z)q⁰(z) +γ3q⁰(z) +tγ4

"

zq⁰⁰(z)

q(z) + q⁰(z) q(z) −z

q⁰(z) q(z)

2#

=a1(t) +. . . and

∂L(z, t)

∂t =γ4

zq⁰(z) q(z) . Also,

∂L(0, t)

∂z =γ₄q⁰(0) γ₃

γ4

+2γ₂ γ4

q(0) +t 1 q(0)

.

From the univalence ofq we have q⁰(0)6= 0 and q(0) = 1, it follows that a1(t)6= 0 for t≥0 and lim

t→∞|a₁(t)|= +∞.

A simple computation yields, Re

( z

∂L(z,t)

∂z

∂L(z,t)

∂t

)

= Re γ₃

γ4

q(z) +2γ₂ γ4

q²(z) +t

1 +zq⁰⁰(z)

q⁰(z) −zq⁰(z)

. Using the fact that q is convex univalent function in U and γ4 6= 0, we have,

Re (

z

∂L(z,t)

∂z

∂L(z,t)

∂t

)

>0 if <

γ3

γ4

q(z) +2γ2

γ4

q²(z)

>0, z∈U, t≥0.

Now, Theorem 4.1 follows by applying Lemma 2.3.

By fixing Φ(z) = _1−z^z and Ψ(z) = _1−z^z in Theorem 4.1, we obtain the following corollary.

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Corollary4.1. Let γi ∈C(i= 1, . . . ,4) (γ4 6= 0), η, α, β∈ C such that η 6= 0 and α+β6= 0,and q be convex univalent withq(0) = 1, and(4.1)hold true. If f ∈A,

αJ_s,b^λ,µ+1f(z)+βJ_s,b^λ,µf(z) (α+β)z

η

∈H[q(0),1]∩Q. Let

γ1+γ2

!^2µ +γ3

!^η

+γ₄η α(µ+ 1)[J_s,b^λ,µ+2f(z)− J_s,b^λ,µ+1f(z)] +βµ[J_s,b^λ,µ+1f(z)− J_s,b^λ,µf(z)]

!

be univalent in Uand

γ1+γ2q²(z) +γ3q(z) +γ4

zq⁰(z) q(z)

≺γ1+γ2

!2µ

+γ3

!η

+γ4η α(µ+ 1)[J_s,b^λ,µ+2f(z)− J_s,b^λ,µ+1f(z)] +βµ[J_s,b^λ,µ+1f(z)− J_s,b^λ,µf(z)]

! ,

then

q(z)≺ αJ_s,b^λ,µ+1f(z) +βJ_s,b^λ,µf(z) (α+β)z

!η

By takings= 0 in Corollary 4.1, we state the following corollary.

Corollary4.2. Let γi ∈C(i= 1, . . . ,4) (γ4 6= 0), η, α, β∈ C such that η 6= 0 and α+β6= 0,and q be convex univalent withq(0) = 1, and(4.1)hold true. If f ∈A,

αI_λ^µ+1f(z)+βI_λ^µf(z) (α+β)z

η

∈H[q(0),1]∩Q. Let γ₁+γ₂q²(z) +γ₃q(z) +γ₄zq⁰(z)

q(z) γ₁ +γ2

!2µ

+γ3

!η

+γ₄η α(µ+ 1)[I_λ^µ+2f(z)− I_λ^µ+1f(z)] +βµ[I_λ^µ+1f(z)− I_λ^µf(z)]

!

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be univalent in Uand

γ1+γ2q²(z) +γ3q(z) +γ4

zq⁰(z) q(z)

≺γ₁+γ₂ αI_λ^µ+1f(z) +βI_λ^µf(z) (α+β)z

!2µ

+γ₃ αI_λ^µ+1f(z) +βI_λ^µf(z) (α+β)z

!η

+γ4η α(µ+ 1)[I_λ^µ+2f(z)− I_λ^µ+1f(z)] +βµ[I_λ^µ+1f(z)− I_λ^µf(z)]

! ,

then

q(z)≺ αI_λ^µ+1f(z) +βI_λ^µf(z) (α+β)z

!η

where I_λ^ηf(z) is the Choi-Saigo-Srivastava operator [9] and q is the best subordinant.

By settings=α= 0, λ= 1, µ= 2 andβ = 1 in Theorem 4.1, we derive the following corollary.

Corollary 4.3. Let Ψ∈A, γ_i ∈C (i= 1, . . . ,4) (γ₄ 6= 0), η∈ C such that η 6= 0 and q be convex univalent with q(0) = 1, and (4.1) hold true. If f ∈A,_(f_∗Ψ)(z)

z

η

∈H[q(0),1]∩Q. Letγ₁+γ₂_(f∗Ψ)(z)

z

2η

+γ₃_(f_∗Ψ)(z)

z

η

+ γ4η

z(f∗Ψ)⁰(z) (f∗Ψ)(z) −1

be univalent in U and

γ1+γ2q²(z) +γ3q(z) +γ4

zq⁰(z) q(z)

≺γ₁+γ₂

(f∗Ψ)(z) z

2η

+γ₃

(f ∗Ψ)(z) z

η

+γ₄η

z(f∗Ψ)⁰(z) (f∗Ψ)(z) −1

then

q(z)≺

(f∗Ψ)(z) z

η

By putting Ψ(z) = _1−z^z ,in Corollary 4.3, we obtain the following corollary.

Corollary 4.4. Let γ_i ∈ C (i= 1, . . . ,4) (γ₄ 6= 0), 0 6= η ∈ C and q be convex univalent with q(0) = 1, and (4.1) hold true. If f ∈ A, _f(z)

z

η

∈ H[q(0),1]∩Q. Letγ1+γ2

f(z) z

2η

+γ3

f(z) z

η

+γ4η zf⁰(z)

f(z) −1

be univalent

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in Uand

γ₁+γ₂q²(z) +γ₃q(z) +γ₄zq⁰(z) q(z)

≺γ₁+γ₂ f(z)

z 2η

+γ₃ f(z)

z η

+γ₄η

zf⁰(z) f(z) −1

then

q(z)≺ f(z)

z η

By takingq(z) = (1 +Az)/(1 +Bz) (−1≤B < A≤1) in Theorem 4.1, we obtain the following corollary.

Corollary4.5. LetΦ,Ψ∈A, γ_i ∈C(i= 1, . . . ,4) (γ₄ 6= 0), η, α, β∈ C such that η6= 0 and α+β 6= 0,and q be convex univalent with q(0) = 1,and Re

γ3

γ4

1+Az 1+Bz

+^2γ_γ²

4

1+Az 1+Bz

2

>0.Iff ∈A,

η

∈H[q(0),1]∩Q. Let ∆^(γⁱ⁾⁴¹(f; Φ,Ψ) be univalent in U and γ1+γ2

1 +Az 1 +Bz

2

+γ3

1 +Az 1 +Bz +γ4

(A−B)z

(1 +Az)(1 +Bz) ≺∆^(γⁱ⁾⁴¹(f; Φ,Ψ), then

1 +Az

1 +Bz ≺ αJ_s,b^λ,µ+1(f∗Φ)(z) +βJ_s,b^λ,µ(f ∗Ψ)(z) (α+β)z

!η

and ^1+Az_1+Bz is the best subordinant.

5. SANDWICH RESULTS

There is a complete analog of Theorem 3.1 for differential subordination and Theorem 4.1 for differential superordination. We can combine the results of Theorem 3.1 with Theorem 4.1 and obtain the following sandwich theorem.

Theorem 5.1. Let q1 and q2 be convex univalent in U, γi ∈ C (i = 1, . . . ,4) (γ₄6= 0), η, α, β∈ C such thatη 6= 0and α+β6= 0,and letq₂ satisfy (3.1)andq₁ satisfy(4.1). Forf,Φ,Ψ∈A,let

η

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∈H[1,1]∩Qand ∆^(γⁱ⁾⁴¹(f; Φ,Ψ) defined by (3.3)be univalent in Usatisfying γ1+γ2q²₁(z) +γ3q1(z) +γ4

zq₁⁰(z) q1(z)

≺∆^(γⁱ⁾⁴¹(f; Φ,Ψ)≺γ1+γ2q²₂(z) +γ3q2(z) +γ4

zq₂⁰(z) q₂(z) , then

q1(z)≺ αJ_s,b^λ,µ+1(f ∗Φ)(z) +βJ_s,b^λ,µ(f∗Ψ)(z) (α+β)z

!^η

≺q2(z) and q₁, q₂ are respectively the best subordinant and best dominant.

By taking q₁(z) = ^1+A_1+B¹^z

1z (−1 ≤ B₁ < A₁ ≤ 1) and q₂(z) = ^1+A_1+B²^z

2z

(−1≤B₂ < A₂ ≤1) in Theorem 5.1 we obtain the following result.

Corollary5.1. Forf,Φ,Ψ∈A,let

η

∈ H[1,1]∩Q and ∆^(γⁱ⁾⁴¹(f; Φ,Ψ) defined by(3.3)be univalent in U satisfying

γ₁+γ₂

1 +A₁z 1 +B1z

2

+γ₃1 +A₁z

1 +B1z +γ₄ (A₁−B₁)z (1 +A1z)(1 +B1z)

≺∆^(γⁱ⁾⁴¹(f; Φ,Ψ)≺γ1+γ2

1 +A2z 1 +B₂z

2

+γ3

1 +A2z 1 +B₂z +γ4

(A2−B2)z (1 +A₂z)(1 +B₂z) then

1 +A1z

1 +B₁z ≺ αJ_s,b^λ,µ+1(f ∗Φ)(z) +βJ_s,b^λ,µ(f∗Ψ)(z) (α+β)z

!^η

≺ 1 +A2z 1 +B₂z and ^1+A_1+B¹^z

1z, _1+B^1+A²^z

2z are respectively the best subordinant and best dominant.

We remark that, one can easily restated Theorem 5.1 for the different choices of Φ(z),Ψ(z), λ, α, β, µ, s, b, γ₁, γ₂, γ₃ and γ₄.

Remark 5.2. (1) Setting µ = 2, β =λ=γ1 = 1, s =γ2 =γ3 =α = 0, Φ(z) = _1−z^z Ψ(z) = _(1−z)^z 2, q(z) = _(1−z)¹2ab (b∈ C − {0}), η =aand γ4 = ¹_b in Theorem 3.1, we get the result obtained by Obradoviˇc et al., [26, Theorem 1].

(2) Setting µ= 2, β =λ=γ₁ = 1, s=γ₂=γ₃ =α= 0,Φ(z) = Ψ(z) =

z

1−z, q(z) = _(1−z)¹ 2b (b∈ C − {0}), η= 1 andγ₄ = ¹_b in Theorem 3.1 and then combining this together with Lemma 2.2, we obtain the result of Srivastava and Lashin [33, Theorem 3].

(3) Takingγ₁ = 1, γ₂ = 0, γ₃ = 0,α= 0, β = 1,Φ(z) = _1−z^z , γ₄ = _ab^e_cos^iλ_λ (a, b∈C,|λ|< ^π₂), η=a and q(z) = (1−z)^−2ab^cos^λe^−iλ in Corollary 3.3, we obtain the result of Aouf et al. [3, Theorem 1].

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(4) For µ = 2, β = λ = 1, s = α = 0, Ψ(z) = z +

∞

P

n=2

[1 + (n− 1)λ]^m^(b)_(c)ⁿ⁻¹

n−1anzⁿ,all the results in [30] are special cases of our results.

(5) Forµ= 2, β=λ= 1, s=α= 0,Ψ(z) =z+

∞

P

n=2

[1 + (n−1)λ]^mb_nzⁿ, all the results in [2] are particular cases of our results.

6. CONCLUSION

We conclude this paper by remarking that in view of the function class defined by the subordination relation (1.16) and expressed in terms of the convolution (1.2) involving arbitrary coefficients, the main results would lead to additional new results. In fact, by appropriately selecting the arbitrary sequences (Φ(z) and Ψ(z)), and specializing the parameters η, λ, b, s, α, β, µ, γ1, γ2, γ3 and γ4 and the function q(z) the results presented in this paper would find further applications for the classes which incorporate generalized forms of linear operators illustrated below:

(1)

Φ(z) =z+

∞

X

n=2

(µ)n−1. . .(αl)n−1

(β1)n−1. . .(βm)n−1

zⁿ (n−1)!

and

Ψ(z) =z+

∞

X

n=2

n (µ)n−1. . .(α_l)n−1

(β1)n−1. . .(βm)n−1

zⁿ (n−1)!,

whereµ, . . . , α_landβ₁, . . . , β_m(l, m∈ N = 1,2,3, . . .) are complex parameters βj ∈ {0,/ −1,−2, . . .}forj= 1,2, . . . m, l≤m+ 1 (results for Dziok-Srivastava operator [11]);

(2) Φ(z) =z+

∞

P

n=2 (a)n−1

(c)n−1

zⁿ

(n−1)! and Ψ(z) =z+

∞

P

n=2

n^(a)_(c)ⁿ⁻¹

n−1

zⁿ

(n−1)!,where c6= 0,−1,−2, . . . (results for Carlson and Shaffer operator [6]);

(3) Φ(z) = _(1−z)^zλ+1, λ ≥ −1, and Ψ(z) = _(1−z)^zλ+2, λ ≥ −1 (results for Ruscheweyh derivative operator [28]);

(4) Φ(z) = z+

∞

P

n=2

n^kzⁿ,and Ψ(z) = z+

∞

P

n=2

n^k+1zⁿ,k ≥0 (results for Salagean operator [29]);

(5) Φ(z) = z+

∞

P

n=2

n+λ 1+λ

k

zⁿ, and Ψ(z) = z+

∞

P

n=2

n n+λ

1+λ

k

zⁿ,where λ≥0,k∈ Z (results for Multiplier transformation [7, 8]),

(16)

in Theorem 3.1, Theorem 4.1 and Theorem 5.1 would eventually lead further new results. These considerations can fruitfully be worked out and we skip the details in this regard.

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Received 10 July 2012 VIT University

School of Advanced Sciences Vellore – 632014, Tamilnadu, India

gmsmoorthy@yahoo.com Government Arts College(Men) P.G. and Research Department of Mathematics

Krishnagiri – 635001, Tamilnadu, India nmagi 2000@yahoo.co.in