SUBORDINATION AND SUPERORDINATION RESULTS FOR Φ-LIKE FUNCTIONS

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Subordination and Superordination T.N. Shanmugam, S. Sivasubramanian and

Maslina Darus vol. 8, iss. 1, art. 20, 2007

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SUBORDINATION AND SUPERORDINATION RESULTS FOR Φ-LIKE FUNCTIONS

T.N. SHANMUGAM S. SIVASUBRAMANIAN

Department of Information Technology, 10/11 Department of Mathematics, Salalah College of Engineering Easwari Engineering College Salalah, Sultanate of Oman Ramapuram, Chennai-600 089, India EMail:drtns2001@yahoo.com EMail:sivasaisastha@rediffmail.com

MASLINA DARUS

School Of Mathematical Sciences, Faculty Of Sciences and Technology, UKM, Malaysia

EMail:maslina@pkrisc.cc.ukm.my

Received: 24 June, 2006

Accepted: 29 December, 2006 Communicated by: N.E. Cho

2000 AMS Sub. Class.: 30C45.

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

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Close Abstract: Letq₁be convex univalent andq₂be univalent in∆ :={z:|z|<1}with

q₁(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 functionfto satisfy

q1(z)≺ z(f ∗g)⁰(z)

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

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

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1. Introduction and Motivations

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

∆ := {z :|z|<1} satisfying f(0) = 0 and f⁰(0) = 1. Let H be the class of functions analytic in∆and for anya ∈Candn ∈N,H[a, n]be the subclass ofH consisting of functions of the formf(z) = a+anzⁿ+an+1zⁿ⁺¹+· · · .Letp, h∈ H and letφ(r, s, t;z) :C³×∆→C. Ifpandφ(p(z), zp⁰²p⁰⁰(z);z)are univalent and if psatisfies the second order superordination

(1.1) h(z)≺φ(p(z), zp⁰²p⁰⁰(z);z),

thenpis a solution of the differential superordination (1.1). Iff is subordinate toF, thenF is called a superordinate off. An analytic functionqis called a subordinant ifq ≺ pfor allpsatisfying (1.1). A univalent subordinantq¯that satisfiesq ≺q¯for all subordinantsq of (1.1) is said to be the best subordinant. Recently Miller and Mocanu [5] obtained conditions on h, q andφ for 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) and q₁(z) ≺ ^z²^f⁰^(z)

{f(z)}² ≺ q₂(z) where q₁ and q₂ are given univalent functions

in ∆with q₁(0) = 1 and q₂(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).

Let Φ be an analytic function in a domain containing f(∆) with Φ(0) = 0 and Φ⁰(0) = 1. For any two analytic functions f(z) = P∞

n=0anzⁿ and g(z) =

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P∞

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

(f ∗g)(z) =

∞

X

n=0

anbnzⁿ.

The functionf ∈ Ais calledΦ-like if

(1.3) <

zf⁰(z) Φ(f(z))

>0 (z ∈∆).

The concept ofΦ−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. LetΦbe analytic in a domain containingf(∆),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 toqby S^∗(q).

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

Definition 1.2. Let g be a fixed function in A.Let Φbe 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

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respect toS_g^∗(q)if

(1.5) z(f∗g)⁰(z)

Φ(f∗g)(z) ≺q(z) (z ∈∆).

We note thatS^∗^z

1−z(q) :=S^∗(q).

In the present investigation, we obtain sufficient conditions for a normalized analytic functionf 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]). LetQbe 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]). Letq be univalent in the open unit disk

∆and θ and φ be analytic in a domainD containing q(∆) with φ(ω) 6= 0 when ω∈q(∆). Setξ(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 ∈∆).

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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 domainDcontainingq(∆). 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.

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2. Main Results

By making use of Lemma1.1, we prove the following result.

Theorem 2.1. Letq(z) 6= 0be analytic and univalent in∆withq(0) = 1such that

zq⁰(z)

q(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) ,

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).

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Then the functionp(z)is analytic in∆withp(0) = 1.By a straightforward compu- tation

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 Theorem2.1, 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 Theorem2.1now follows by an application of Lemma1.1.

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WhenΦ(ω) =ωin Theorem2.1we 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.

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 = 1andq(z) = ^1+Az_1+Bz (−1≤B < A≤1)in Corollary2.3, 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),

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then zf⁰(z)

f(z) ≺ 1 +Az 1 +Bz and ^1+Az_1+Bz 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∗g)_Φ(f∗g)(z)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 Theorem2.5follows by an application of Lemma1.2.

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ForΦ(ω) = ωin Theorem2.5, 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.

Combining Theorem2.1 and Theorem 2.5we 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 that q₁(0) = 1, q₂(0) = 1, ^zq_q¹⁰^(z)

1(z) and ^zq_q⁰²^(z)

2(z) are starlike univalent in∆with

q₁(z)6= 0 and q₂(z)6= 0.

Letf ∈ 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) + γzq₂⁰(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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References

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[7] St. RUSCHEWEYH, A subordination theorem forF-like functions, J. London Math. Soc., 13(2) (1976), 275–280.

[8] T.N. SHANMUGAM, V. RAVICHANDRANANDS. SIVASUBRAMANIAN, Differential sandwich theorems for some subclasses of analytic Functions, Aust. J. Math. Anal. Appl., 3(1) (2006), Art. 8, 11 pp. (electronic).

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[10] T.N. SHANMUGAM, S. SIVASUBRAMANIAN AND M. DARUS, On cer- tain subclasses of functions involving a linear Operator, Far East J. Math. Sci., 23(3), (2006), 329–339.

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