Received24June,2006;accepted29December,2006CommunicatedbyN.E.Cho andforany a ∈ C and n ∈ N , H [ a,n ] bethesubclassof H consistingoffunctionsoftheform { z : | z | < 1 } satisfying f (0)=0 and f (0)=1 . Let H betheclassoffunctionsanalyticin ∆ Let A bethec

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SUBORDINATION AND SUPERORDINATION RESULTS FOR Φ-LIKE FUNCTIONS

T.N. SHANMUGAM, S. SIVASUBRAMANIAN, AND MASLINA DARUS DEPARTMENT OFINFORMATIONTECHNOLOGY,

SALALAHCOLLEGE OFENGINEERING

SALALAH, SULTANATE OFOMAN

drtns2001@yahoo.com DEPARTMENT OFMATHEMATICS, EASWARIENGINEERINGCOLLEGE

RAMAPURAM, CHENNAI-600 089 INDIA

sivasaisastha@rediffmail.com SCHOOLOFMATHEMATICALSCIENCES, FACULTYOFSCIENCES ANDTECHNOLOGY,

UKM, MALAYSIA

maslina@pkrisc.cc.ukm.my

Received 24 June, 2006; accepted 29 December, 2006 Communicated by N.E. Cho

ABSTRACT. Letq1be convex univalent andq2be univalent in∆ :={z:|z|<1}withq1(0) = q2(0) = 1. Letf be a normalized analytic function in the open unit disk∆. LetΦbe an analytic function in a domain containingf(∆),Φ(0) = 0andΦ⁰(0) = 1. We give some applications of first order differential subordination and superordination to obtain sufficient conditions for the functionf to satisfy

q1(z)≺z(f∗g)⁰(z)

Φ(f∗g)(z) ≺q2(z) wheregis a fixed function.

Key words and phrases: Differential subordination, Differential superordination, Convolution, Subordinant.

2000 Mathematics Subject Classification. 30C45.

1. INTRODUCTION ANDMOTIVATIONS

Let A be the class of all normalized analytic functions f(z) in the open unit disk ∆ :=

{z :|z|<1}satisfyingf(0) = 0andf⁰(0) = 1.LetHbe the class of functions analytic in∆ and for anya ∈Candn ∈N,H[a, n]be the subclass ofHconsisting of functions of the form

We would like to thank the referee for his insightful suggestions.

175-06

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f(z) =a+a_nzⁿ+a_n+1zⁿ⁺¹+· · · .Letp, h∈ Hand letφ(r, s, t;z) :C³×∆→C. Ifpand φ(p(z), zp⁰²p⁰⁰(z);z)are univalent and ifpsatisfies the second order superordination

(1.1) h(z)≺φ(p(z), zp⁰²p⁰⁰(z);z),

then pis a solution of the differential superordination (1.1). If f is subordinate to F, then F is called a superordinate off. An analytic functionq is called a subordinant ifq ≺ pfor all p satisfying (1.1). A univalent subordinant q¯that satisfies q ≺ q¯for all subordinantsq of (1.1) is said to be the best subordinant. Recently Miller and Mocanu [5] obtained conditions onh, q andφfor which the following implication holds:

(1.2) h(z)≺φ(p(z), zp⁰²p⁰⁰(z);z)⇒q(z)≺p(z).

Using the results of Miller and Mocanu [4], Bulboac˘a [2] considered certain classes of first order differential superordinations as well as superordination-preserving integral operators [1].

In an earlier investigation, Shanmugam et al. [8] obtained sufficient conditions for a normalized analytic functionf(z)to satisfyq₁(z) ≺ _zf^f(z)0(z) ≺ q₂(z)andq₁(z) ≺ ^z²^f⁰^(z)

{f(z)}² ≺ q₂(z)whereq₁

andq₂ are given univalent functions in∆withq₁(0) = 1andq₂(0) = 1.A systematic study of the subordination and superordination has been studied very recently by Shanmugam et al. in [9] and [10] (see also the references cited by them).

LetΦbe an analytic function in a domain containingf(∆)withΦ(0) = 0andΦ⁰(0) = 1. For any two analytic functionsf(z) = P∞

n=0a_nzⁿandg(z) = P∞

n=0b_nzⁿ,the Hadamard product or convolution off(z)andg(z), written as(f ∗g)(z)is defined by

(f ∗g)(z) =

∞

X

n=0

a_nb_nzⁿ.

The functionf ∈ Ais calledΦ-like if

(1.3) <

zf⁰(z) Φ(f(z))

>0 (z ∈∆).

The concept of Φ− like functions was introduced by Brickman [3] and he established that a functionf ∈ Ais univalent if and only iff isΦ-like for someΦ.ForΦ(w) = w,the functionf is starlike. In a later investigation, Ruscheweyh [7] introduced and studied the following more general class ofΦ-like functions.

Definition 1.1. Let Φ be analytic in a domain containing f(∆), Φ(0) = 0, Φ⁰(0) = 1 and Φ(w) 6= 0 for w ∈ f(∆)\ {0}.Let q(z)be a fixed analytic function in ∆, q(0) = 1.The functionf ∈ Ais calledΦ-like with respect toqif

(1.4) zf⁰(z)

Φ(f(z)) ≺q(z) (z ∈∆).

WhenΦ(w) =w,we denote the class of allΦ-like functions with respect toqbyS^∗(q).

Using the definition ofΦ−like functions, we introduce the following class of functions.

Definition 1.2. Letgbe a fixed function inA.LetΦbe analytic in a domain containingf(∆), Φ(0) = 0, Φ⁰(0) = 1andΦ(w)6= 0forw∈ f(∆)\ {0}.Letq(z)be a fixed analytic function in∆,q(0) = 1.The functionf ∈ Ais calledΦ-like with respect toS_g^∗(q)if

(1.5) z(f ∗g)⁰(z)

Φ(f ∗g)(z) ≺q(z) (z ∈∆).

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We note thatS^∗z

1−z(q) :=S^∗(q).

In the present investigation, we obtain sufficient conditions for a normalized analytic function f to satisfy

q₁(z)≺ z(f∗g)⁰(z)

Φ(f ∗g)(z) ≺q₂(z).

We shall need the following definition and results to prove our main results. In this sequel, unless otherwise stated,αandγ are complex numbers.

Definition 1.3 ([4, Definition 2, p. 817]). Let Qbe the set of all functionsf 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 1.1 ([4, Theorem 3.4h, p. 132]). Let q be univalent in the open unit disk ∆ and θ and φ be analytic in a domain D containing q(∆) with φ(ω) 6= 0 when ω ∈ q(∆). Set ξ(z) = zq⁰(z)φ(q(z)),h(z) =θ(q(z)) +ξ(z). Suppose that

(1) ξ(z)is starlike univalent in∆, and (2) <n_zh0(z)

ξ(z)

o

>0 (z ∈∆).

Ifpis analytic in∆withp(∆)⊆Dand

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

Lemma 1.2. [2, Corollary 3.1, p. 288] Letqbe univalent in∆,ϑandϕbe analytic in a domain Dcontainingq(∆). Suppose that

(1) <h

ϑ⁰(q(z)) ϕ(q(z))

i

>0forz ∈∆, and

(2) ξ(z) =zq⁰(z)ϕ(q(z))is starlike univalent function in∆.

Ifp∈ H[q(0),1]∩Q,withp(∆)⊂D,andϑ(p(z)) +zp⁰(z)ϕ(p(z))is univalent in∆,and (1.7) ϑ(q(z)) +zq⁰(z)ϕ(q(z))≺ϑ(p(z)) +zp⁰(z)ϕ(p(z)),

thenq(z)≺p(z)andqis the best subordinant.

2. MAINRESULTS

By making use of Lemma 1.1, we prove the following result.

Theorem 2.1. Let q(z) 6= 0 be analytic and univalent in ∆with q(0) = 1 such that ^zq_q(z)⁰^(z) is starlike univalent in∆.Letq(z)satisfy

(2.1) <

1 + αq(z)

γ − zq⁰(z)

q(z) +zq⁰⁰(z) q⁰(z)

>0.

Let

(2.2) Ψ(α, γ, g;z) := α

z(f∗g)⁰(z) Φ(f∗g)(z)

+γ

1 + z(f ∗g)⁰⁰(z)

(f∗g)⁰(z) − z(Φ(f∗g)(z))⁰ Φ(f∗g)(z)

.

Ifqsatisfies

(2.3) Ψ(α, γ, g;z)≺αq(z) + γzq⁰(z)

q(z) ,

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then

z(f ∗g)⁰(z)

Φ(f ∗g)(z) ≺q(z) andqis the best dominant.

Proof. Let the functionp(z)be defined by

(2.4) p(z) := z(f ∗g)⁰(z)

Φ(f ∗g)(z).

Then the functionp(z)is analytic in∆withp(0) = 1.By a straightforward computation zp⁰(z)

p(z) =

1 + z(f ∗g)⁰⁰(z)

(f∗g)⁰(z) − z[Φ(f∗g)(z)]⁰ Φ(f ∗g)(z)

which, in light of hypothesis (2.3) of Theorem 2.1, yields the following subordination

(2.5) αp(z) + γzp⁰(z)

p(z) ≺αq(z) + γzq⁰(z) q(z) . By setting

θ(ω) :=αω and φ(ω) := γ ω,

it can be easily observed thatθ(ω)andφ(ω)are analytic inC\ {0}and that φ(ω)6= 0 (ω∈C\ {0}).

Also, by letting

(2.6) ξ(z) =zq⁰(z)φ(q(z)) = γ

q(z)zq⁰(z).

and

(2.7) h(z) =θ{q(z)}+ξ(z) = αq(z) + γ

q(z)zq⁰(z), we find thatξ(z)is starlike univalent in∆and that

<

1 + αq(z)

γ − zq⁰(z)

q(z) + zq⁰⁰(z) q⁰(z)

>0

by the hypothesis (2.1). The assertion of Theorem 2.1 now follows by an application of Lemma

1.1.

WhenΦ(ω) = ωin Theorem 2.1 we get:

Corollary 2.2. Letq(z)6= 0be univalent in∆withq(0) = 1. Ifqsatisfies

(α−γ)z(f∗g)⁰(z) (f ∗g)(z) +γ

1 + z(f∗g)⁰⁰(z) (f ∗g)⁰(z)

≺αq(z) + γzq⁰(z) q(z) , then

z(f ∗g)⁰(z)

(f ∗g)(z) ≺q(z) andqis the best dominant.

Forg(z) = _1−z^z andΦ(ω) =ω,we get the following corollary.

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Corollary 2.3. Letq(z)6= 0be univalent in∆withq(0) = 1. Ifqsatisfies

(α−γ)zf⁰(z) f(z) +γ

1 + zf⁰⁰(z) f⁰(z)

≺αq(z) + γzq⁰(z) q(z) ,

then zf⁰(z)

f(z) ≺q(z) andqis the best dominant.

For the choiceα =γ = 1 andq(z) = ^1+Az_1+Bz (−1 ≤ B < A ≤ 1)in Corollary 2.3, we have the following result of Ravichandran and Jayamala [6].

Corollary 2.4. Iff ∈ Aand

1 + zf⁰⁰(z)

f⁰(z) ≺ 1 +Az

1 +Bz + (A−B)z (1 +Az)(1 +Bz),

then zf⁰(z)

f(z) ≺ 1 +Az 1 +Bz and _1+Bz^1+Az is the best dominant.

Theorem 2.5. Letγ 6= 0. Letq(z) 6= 0be convex univalent in∆withq(0) = 1such that ^zq_q(z)⁰^(z) is starlike univalent in∆.Suppose thatq(z)satisfies

(2.8) <

αq(z) γ

>0.

Iff ∈ A, ^z(f_Φ(f∗g)(z)^∗g)⁰^(z) ∈ H[1,1]∩Q,Ψ(α, γ, g;z)as defined by (2.2) is univalent in∆and

(2.9) αq(z) + γzq⁰(z)

q(z) ≺Ψ(α, γ, g;z), then

q(z)≺ z(f ∗g)⁰(z) Φ(f ∗g)(z) andqis the best subordinant.

Proof. By setting

ϑ(w) :=αω and ϕ(w) := γ ω,

it is easily observed thatϑ(w)is analytic inC,ϕ(w)is analytic inC\ {0}and that ϕ(w)6= 0, (w∈C\ {0}).

The assertion of Theorem 2.5 follows by an application of Lemma 1.2.

ForΦ(ω) = ωin Theorem 2.5, we get

Corollary 2.6. Letq(z)6= 0be convex univalent in∆withq(0) = 1. Iff ∈ Aand

αq(z) + γzq⁰(z)

q(z) ≺(α−γ)

z(f ∗g)⁰(z) (f∗g)(z)

+γ

1 + z(f∗g)⁰⁰(z) (f∗g)⁰(z)

,

then

q(z)≺ z(f ∗g)⁰(z) (f ∗g)(z) andqis the best subordinant.

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Combining Theorem 2.1 and Theorem 2.5 we get the following sandwich theorem.

Theorem 2.7. Let q1 be convex univalent andq2 be univalent in ∆ satisfying (2.8) and (2.1) respectively such thatq₁(0) = 1, q₂(0) = 1, ^zq_q⁰¹^(z)

1(z) and ^zq_q⁰²^(z)

2(z) are starlike univalent in∆with q₁(z)6= 0 and q₂(z)6= 0.

Let f ∈ A, ^z(f_Φ(f∗g)(z)^∗g)⁰^(z) ∈ H[1,1]∩Q, andΨ(α, γ, g;z) as defined by (2.2) be univalent in ∆.

Further, if

αq₁(z) + γzq₁⁰(z)

q1(z) ≺Ψ(α, γ, g;z)≺αq₂(z) + γzq⁰₂(z) q2(z) , then

q₁(z)≺ z(f ∗g)⁰(z)

Φ(f∗g)(z) ≺q₂(z)

andq₁ andq₂are respectively the best subordinant and best dominant.

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[3] L. BRICKMAN,Φ-like analytic functions. I, Bull. Amer. Math. Soc., 79 (1973), 555–558.

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