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volume 7, issue 4, article 152, 2006.

Received 27 March, 2006;

accepted 19 September, 2006.

Communicated by:N.K. Govil

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Journal of Inequalities in Pure and Applied Mathematics

DIFFERENTIAL SUBORDINATIONS AND SUPERORDINATIONS FOR ANALYTIC FUNCTIONS DEFINED BY THE DZIOK-SRIVASTAVA LINEAR OPERATOR

G. MURUGUSUNDARAMOORTHY AND N. MAGESH

School of Science and Humanities Vellore Institute of Technology

Deemed University, Vellore - 632014, India.

EMail:gmsmoorthy@yahoo.com Department of Mathematics Adhiyamaan College of Engineering Hosur - 635109, India.

EMail:nmagi_2000@yahoo.co.in

2000c Victoria University ISSN (electronic): 1443-5756 092-06

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Differential Subordinations and Superordinations for Analytic

Functions Defined by the Dziok-Srivastava Linear

Operator

G. Murugusundaramoorthy and N. Magesh

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Abstract

In the present investigation, we obtain some subordination and superordination results involving Dziok-Srivastava linear operatorH_m^l [α₁]for certain normalized analytic functions in the open unit disk. Our results extend corresponding pre- viously known results.

2000 Mathematics Subject Classification:Primary 30C45; Secondary 30C80.

Key words: Univalent functions, Starlike functions, Convex functions, Differential subordination, Convolution, Dziok-Srivastava linear operator.

1. Introduction

Let H be the class of functions analytic in ∆ := {z : |z| < 1} and H(a, n) be the subclass of H consisting of functions of the form f(z) = a+anzⁿ + a_n+1zⁿ⁺¹+· · ·. LetAbe the subclass ofHconsisting of functions of the form f(z) = z+a₂z²+· · ·. Letp, h∈ H and letφ(r, s, t;z) : C³×∆→ C. Ifp and φ(p(z), zp⁰(z), z²p⁰⁰(z);z)are univalent and ifpsatisfies the second order superordination

(1.1) h(z)≺φ(p(z), zp⁰(z), z²p⁰⁰(z);z),

thenpis a solution of the differential superordination (1.1). (Iff is subordinate toF, thenF is superordinate tof.) An analytic functionqis called a subordi- nant if q ≺ pfor allpsatisfying (1.1). A univalent subordinantqethat satisfies q ≺ eq for all subordinants q of (1.1) is said to be the best subordinant. Re- cently Miller and Mocanu [14] obtained conditions onh,qandφfor which the following implication holds:

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

Using the results of Miller and Mocanu [14], Bulboac˘a [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 Bulboac˘a [5] and obtained sufficient conditions for certain normalized analytic functions f(z)to satisfy

q₁(z)≺ zf⁰(z)

f(z) ≺q₂(z),

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whereq₁andq₂ are given univalent functions in∆withq₁(0) = 1andq₂(0) = 1.Shanmugam et al. [19] obtained sufficient conditions for a normalized analytic functionf(z)to satisfy

q₁(z)≺ f(z)

zf⁰(z) ≺q₂(z) and q₁(z)≺ z²f⁰(z)

{f(z)}² ≺q₂(z)

whereq₁andq₂ are given univalent functions in∆withq₁(0) = 1andq₂(0) = 1.

In [2], for functionsf ∈ Asuch thatδ >0,

<

(zf⁰(z) f(z)

f(z) z

δ)

>0, z ∈∆,

a class of Bazilevic type functions was considered and certain properties were studied. In this paper motivated by Liu [11], we define a class

B(λ, δ, A, B) :=

(

f ∈ A: (1−λ)

f(z) z

δ

+λzf⁰(z) f(z)

f(z) z

δ

≺ 1 +Az 1 +Bz

) ,

where δ > 0, λ ≥ 0, −1 ≤ B < A ≤ 1 and studied certain interesting properties based on subordination. Further we obtained a sandwich result for functions in the classB(λ, δ, A, B).

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2. Preliminaries

For our present investigation, we shall need the following definition and results.

Definition 2.1 ([14, Definition 2, p. 817]). Denote byQ, the set of all functions f(z)that are analytic and injective on∆−E(f), where

E(f) =

ζ ∈∂∆ : lim

z→ζf(z) = ∞

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

Lemma 2.1 ([13, Theorem 3.4h, p. 132]). Let q(z) be univalent in the unit disk∆andθandφbe analytic in a domainDcontainingq(∆)withφ(w)6= 0 whenw ∈ q(∆). SetQ(z) = zq⁰(z)φ(q(z)), h(z) = θ(q(z)) +Q(z). Suppose that

1. Q(z)is starlike univalent in∆, and 2. <n

zh⁰(z) Q(z)

o

>0forz∈∆.

If

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

Lemma 2.2 ([19]). Letqbe a convex univalent function in∆andψ, γ ∈Cwith

<

1 + zq⁰⁰(z) q⁰(z) + ψ

γ

>0.

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Ifp(z)is analytic in∆and

ψp(z) +γzp⁰(z)≺ψq(z) +γzq⁰(z) thenp(z)≺q(z)andq(z)is the best dominant.

Lemma 2.3 ([5]). Letq(z)be convex univalent in the unit disk∆andϑandϕ be analytic in a domainDcontainingq(∆). Suppose that

1. <[ϑ⁰(q(z))/ϕ(q(z))]>0forz ∈∆, 2. zq⁰(z)ϕ(q(z))is starlike univalent in∆.

Ifp(z) ∈ H[q(0),1]∩Q, withp(∆) ⊆ D, and ϑ(p(z)) +zp⁰(z)ϕ(p(z))is univalent in∆, and

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

Lemma 2.4 ([14, Theorem 8, p. 822]). Let q be convex univalent in ∆ and γ ∈C. Further assume that<[γ]>0.Ifp(z)∈ H[q(0),1]∩Q,p(z) +γzp⁰(z) is univalent in∆, then

q(z) +γzq⁰(z)≺p(z) +γzp⁰(z) impliesq(z)≺p(z)andq(z)is the best subordinant.

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For two functionsf(z) = z +P∞

n=2a_nzⁿ andg(z) = z+P∞

n=2b_nzⁿ, the Hadamard product (or convolution) off andg is defined by

(f∗g)(z) :=z+

∞

X

n=2

a_nb_nzⁿ =: (g∗f)(z).

Forαj ∈C (j = 1,2, . . . , l)andβj ∈C\ {0,−1,−2, . . .}(j = 1,2, . . . , m), the generalized hypergeometric function _lF_m(α₁, . . . , α_l;β₁, . . . , β_m;z) is defined by the infinite series

lFm(α1, . . . , αl;β1, . . . , βm;z) :=

∞

X

n=0

(α₁)_n· · ·(α_l)_n (β₁)_n· · ·(β_m)_n

zⁿ n!

(l ≤m+ 1;l, m∈N0 :={0,1,2, . . .}) where(a)_nis the Pochhammer symbol defined by

(a)_n:= Γ(a+n) Γ(a)

=

( 1, (n= 0);

a(a+ 1)(a+ 2)· · ·(a+n−1), (n∈N:={1,2,3. . .}).

Corresponding to the function

h(α₁, . . . , α_l;β₁, . . . , β_m;z) :=z_lF_m(α₁, . . . , α_l;β₁, . . . , β_m;z),

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the Dziok-Srivastava operator [7] (see also [8,20])H_m^l (α₁, . . . , α_l;β₁, . . . , β_m) is defined by the Hadamard product

H_m^l (α₁, . . . , α_l;β₁, . . . , β_m)f(z) (2.2)

:=h(α₁, . . . , α_l;β₁, . . . , β_m;z)∗f(z)

=z+

∞

X

n=2

(α₁)n−1· · ·(α_l)n−1

(β1)n−1· · ·(βm)n−1

a_nzⁿ (n−1)!. For brevity, we write

H_m^l [α₁]f(z) :=H_m^l (α₁, . . . , α_l;β₁, . . . , β_m)f(z).

It is easy to verify from (2.2) that

(2.3) z(H_m^l [α₁]f(z))⁰ =α₁H_m^l [α₁+ 1]f(z)−(α₁−1)H_m^l [α₁]f(z).

Special cases of the Dziok-Srivastava linear operator include the Hohlov linear operator [9], the Carlson-Shaffer linear operatorL(a, c)[6], the Ruscheweyh derivative operatorDⁿ [18], the generalized Bernardi-Libera-Livingston linear integral operator (cf. [3], [10], [12]) and the Srivastava-Owa fractional deriva- tive operators (cf. [16], [17]).

The main object of the present paper is to find sufficient conditions for certain normalized analytic functionsf(z)to satisfy

q₁(z)≺

H_m^l [α₁]f(z) z

^δ

≺q₂(z)

whereq₁ andq₂are given univalent functions in∆.Also, we obtain the number of known results as special cases.

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3. Main Results

We begin with the following:

Theorem 3.1. Letq(z)be univalent in∆, λ ∈C andα1 >0, δ >0.Suppose q(z)satisfies

(3.1) <

1 + zq⁰⁰(z) q⁰(z) + λ

δ

>0.

Iff ∈ Asatisfies the subordination,

(3.2) (1−λα₁)

H_m^l [α₁]f(z) z

^δ

+λα₁

H_m^l [α1]f(z) z

δ

H_m^l [α1 + 1]f(z) H_m^l [α₁]f(z)

≺q(z) + λ

δzq⁰(z), then

H_m^l [α₁]f(z) z

δ

≺q(z)

andq(z)is the best dominant.

Proof. Define the functionp(z)by

(3.3) p(z) :=

H_m^l [α₁]f(z) z

δ

.

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Then

zp⁰(z) δ :=α₁

H_m^l [α₁]f(z) z

^δ

H_m^l [α₁+ 1]f(z) H_m^l [α₁]f(z) −1

,

hence the hypothesis (3.2) of Theorem3.1yields the subordination:

p(z) + λzp⁰(z)

δ ≺q(z) + λzq⁰(z) δ .

Now Theorem3.1follows by applying Lemma2.2withψ = 1andγ = ^λ_δ. Whenl = 2, m = 1, α₁ =a, α₂ = 1,andβ₁ = cin Theorem3.1, we have the following corollary.

Corollary 3.2. Let q(z)be univalent in∆, λ ∈Candα₁ >0, δ >0.Suppose q(z)satisfies (3.1). Iff ∈ Aand satisfies the subordination,

(3.4) (1−λa)

L(a, c)f(z) z

δ

+λa

L(a, c)f(z) z

δ

L(a+ 1, c)f(z) L(a, c)f(z)

≺q(z) + λ

δzq⁰(z), then

L(a, c)f(z) z

δ

≺q(z)

By takingl = 1, m = 0andα₁ = 1in Theorem3.1, we get the following corollary.

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Corollary 3.3. Let q(z)be univalent in∆, λ ∈Candα₁ >0, δ >0.Suppose q(z)satisfies (3.1). Iff ∈ Aand satisfies the subordination,

(3.5) (1−λ)

f(z) z

δ

+λ

f(z) z

δ zf⁰(z)

f(z)

≺q(z) + λ

δzq⁰(z), then

f(z) z

δ

≺q(z)

Corollary 3.4. Let−1≤B < A≤1and (3.1) hold. Iff ∈ Aand

(1−λα₁)

H_m^l [α₁]f(z) z

δ

+λα₁

H_m^l [α₁]f(z) z

δ

H_m^l [α₁+ 1]f(z) H_m^l [α₁]f(z)

≺ λ(A−B)z

δ(1 +Bz)² + 1 +Az 1 +Bz, then

H_m^l [α₁]f(z) z

^δ

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

Theorem 3.5. Letq(z)be univalent in∆,λ, δ ∈C.Supposeq(z)satisfies

(3.6) <

1 + zq⁰⁰(z)

q⁰(z) − zq⁰(z) q(z)

>0.

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Iff ∈ Asatisfies the subordination:

(3.7) 1 +γδα₁

H_m^l [α₁+ 1]f(z) H_m^l [α1]f(z) −1

≺1 +γzq⁰(z) q(z) , then

H_m^l [α₁]f(z) z

^δ

≺q(z)

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

H_m^l [α₁]f(z) z

δ

.

It is clear that p(0) = 1andp(z)is analytic in∆. By using the identity (2.3), from (3.3) we get,

(3.8) zp⁰(z)

p(z) =α₁δ

H_m^l [α₁+ 1]f(z) H_m^l [α1]f(z) −1

.

Using (3.8) in (3.7), we see that the subordination becomes 1 +γzp⁰(z)

p(z) ≺1 +γzq⁰(z) q(z) . By setting

θ(w) = 1 and ϕ(w) = γ w,

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we observe thatϕandθare analytic inC\ {0}. Also we see that Q(z) :=zq⁰(z)ϕ(q(z)) = γzq⁰(z)

q(z) , and

h(z) := ϑ(q(z)) +Q(z) = 1 +γzq⁰(z) q(z) . It is clear thatQ(z)is starlike univalent in∆and

<zh⁰(z) Q(z) =<

1 + zq⁰⁰(z)

q⁰(z) − zq⁰(z) q(z)

≥0.

By the hypothesis of Theorem 3.5, the result now follows by an application of Lemma2.1.

Specializing the values ofl = 1, m = 0, α₁ = 1 and q(z) = _(1−z)¹ 2b (b ∈ C − {0}), γ = ¹_b and δ = 1 in Theorem 3.5 above, we have the following corollary as stated in [21].

Corollary 3.6. Letbbe a non zero complex number. Iff ∈ Aand 1 + 1

b

zf⁰(z) f(z) −1

≺ 1 +z 1−z,

then f(z)

z ≺ 1

(1−z)^2b and _(1−z)¹ 2b is the best dominant.

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Choosing the values of l = 1, m = 0, α₁ = 1 and q(z) = _(1−z)¹2ab (b ∈ C− {0}), γ = ¹_b andδ =a 6= 0in Theorem3.5above, we have the following corollary as stated in [15].

b

zf⁰(z) f(z) −1

≺ 1 +z 1−z,

then

f(z) z

a

≺ 1

(1−z)^2ab

wherea6= 0is a complex number and _(1−z)¹2ab is the best dominant.

Similarly for l = 2, m = 1, α₁ = 1, α₂ = 1, β₁ = 1and q(z) = _(1−z)¹ 2b

(b ∈ C− {0}), γ = ¹_b andδ = 1in Theorem3.5 above, we get the following result as stated in [21].

b

zf⁰⁰(z) f⁰(z) −1

≺ 1 +z 1−z, then

f⁰(z)≺ 1 (1−z)^2b and _(1−z)¹ 2b is the best dominant.

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Next, applying Lemma2.3, we have the following theorem.

Theorem 3.9. Let q(z) be convex univalent in ∆, λ ∈ C and 0 < δ < 1.

Supposef ∈ Asatisfies

(3.9) Re

δ λ

>0

and

H_m^l[α1]f(z) z

δ

∈H[q(0),1]∩Q.Let

(1−λα1)

H_m^l [α₁]f(z) z

^δ +λα1

H_m^l [α₁]f(z) z

^δ

H_m^l [α₁+ 1]f(z) H_m^l [α₁]f(z)

be univalent in∆.Iff ∈ Asatisfies the superordination,

(3.10) q(z) + λ

δzq⁰(z)≺(1−λα1)

H_m^l [α₁]f(z) z

^δ

+λα₁

H_m^l [α₁]f(z) z

^δ

H_m^l [α₁+ 1]f(z) H_m^l [α₁]f(z)

then

q(z)≺

H_m^l [α₁]f(z) z

δ

andq(z)is the best subordinant.

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Proof. Define the functionp(z)by

(3.11) p(z) :=

H_m^l [α₁]f(z) z

^δ .

Using (3.11), simple computation produces zp⁰(z)

δ :=α₁

H_m^l [α1]f(z) z

δ

H_m^l [α1+ 1]f(z) H_m^l [α₁]f(z) −1

, then

q(z) + λ

δzq⁰(z)≺p(z) + λ

δzp⁰(z).

By setting ϑ(w) = w and φ(w) = ^λ_δ, it is easily observed that ϑ(w) is analytic inC.Also,φ(w)is analytic inC\{0}andφ(w)6= 0,(w∈C\{0}).

Sinceq(z)is a convex univalent function, it follows that

<

ϑ⁰(q(z)) φ(q(z))

=<

δ λ

>0, z ∈∆, δ, λ∈C, δ, λ6= 0.

Now Theorem3.9follows by applying Lemma2.3.

Concluding the results of differential subordination and superordination, we state the following sandwich result.

Theorem 3.10. Let q₁ andq₂ be convex univalent in∆, λ∈ Cand0< δ < 1.

Supposeq₂satisfies (3.1) andq₁ satisfies (3.9). If

H_m^l [α1]f(z) z

δ

∈ H[q(0),1]∩

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Q,

(1−λα₁)

H_m^l [α₁]f(z) z

^δ +λα₁

H_m^l [α₁]f(z) z

^δ

H_m^l [α₁+ 1]f(z) H_m^l [α1]f(z)

is univalent in∆.Iff ∈ Asatisfies

q₁(z) + λ

δzq₁⁰(z)≺(1−λα₁)

H_m^l [α₁]f(z) z

δ

(3.12)

+λα₁

H_m^l [α₁]f(z) z

^δ

H_m^l [α₁+ 1]f(z) H_m^l [α₁]f(z)

≺q₂(z) + λ

δzq⁰₂(z), then

q₁(z)≺

H_m^l [α₁]f(z) z

^δ

≺q₂(z)

andq₁, q₂are respectively the best subordinant and best dominant.

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[10] R.J. LIBERA, Some classes of regular univalent functions, Proc. Amer.

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[20] 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 (2001), 209–243.

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