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volume 7, issue 4, article 134, 2006.

Received 21 June, 2005;

accepted 01 June, 2006.

Communicated by:A. Sofo

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Journal of Inequalities in Pure and Applied Mathematics

SUBORDINATION RESULTS FOR A CLASS OF ANALYTIC FUNCTIONS DEFINED BY A LINEAR OPERATOR

B.A. FRASIN

Department of Mathematics Al al-Bayt University P.O. Box: 130095 Mafraq, Jordan

EMail:[email protected]

2000c Victoria University ISSN (electronic): 1443-5756 189-05

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Subordination Results for a Class of Analytic Functions Defined by a Linear Operator

B.A. Frasin

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J. Ineq. Pure and Appl. Math. 7(4) Art. 134, 2006

http://jipam.vu.edu.au

Abstract

In this paper, we derive several interesting subordination results for certain class of analytic functions defined by the linear operatorL(a, c)f(z)which introduced and studied by Carlson and Shaffer [2].

2000 Mathematics Subject Classification: Primary 30C45; Secondary 30A10, 30C80.

Key words: Analytic functions, Hadamard product, Subordinating factor sequence.

1. Introduction and Definitions

LetAdenote the class of functions of the form:

(1.1) f(z) =z+

∞

X

n=2

anzⁿ

which are analytic in the open unit disc ∆ = {z :|z|<1}.For two functions f(z)andg(z)given by

(1.2) f(z) =z+

∞

X

n=2

a_nzⁿ and g(z) = z+

∞

X

n=2

c_nzⁿ

their Hadamard product (or convolution) is defined by

(1.3) (f ∗g)(z) :=z+

∞

X

n=2

ancnzⁿ.

Define the functionφ(a, c;z)by (1.4) φ(a, c;z) :=

∞

X

n=0

(a)_n

(c)_nzⁿ⁺¹ (c /∈Z⁻0 :={0,−1,−2, . . .}, z∈∆), where(λ)_nis the Pochhammer symbol given, in terms of Gamma functions,

(λ)_n:= Γ(λ+n) (1.5) Γ(λ)

=

( 1, n= 0,

λ(λ+ 1)(λ+ 2). . .(λ+n−1), n∈N:{1,2, . . .}.

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Corresponding to the functionφ(a, c;z),Carlson and Shaffer [2] introduced a linear operatorL(a, c) :A → Aby

(1.6) L(a, c)f(z) := φ(a, c;z)∗f(z), or, equivalently, by

L(a, c)f(z) :=z+

∞

X

n=1

(a)_n

(c)_na_n+1zⁿ⁺¹ (z ∈∆).

Note thatL(1,1)f(z) =f(z),L(2,1)f(z) =zf⁰(z)andL(3,1)f(z) =zf⁰(z)+

1

2z²f⁰⁰(z).

For−1 ≤ α < 1, β ≥ 0, we let L(a, c;α, β) consist of functions f in A satisfying the condition

(1.7) Re

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

L(a, c)f(z) −(a−1)

> β

aL(a+ 1, c)f(z) L(a, c)f(z) −a

+α, (z ∈∆) The family L(a, c;α, β) is of special interest for it contains many well- known as well as many new classes of analytic univalent functions. ForL(1,1;α,0), we obtain the family of starlike functions of orderα(0≤α <1)andL(2,1;α,0) is the family of convex functions of orderα(0≤ α <1).ForL(1,1; 0, β)and L(2,1; 0, β), we obtain the class of uniformly β- starlike functions and uni- formlyβ- convex functions, respectively, introduced by Kanas and Winsiowska

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([3],[4]) (see also the work of Kanas and Srivastava [5], Goodman ([7],[8]), Rønning ([10],[11]), Ma and Minda [9] and Gangadharan et al. [6]).

Before we state and prove our main result we need the following definitions and lemmas.

Definition 1.1 (Subordination Principle). Let g(z)be analytic and univalent in∆.Iff(z)is analytic in∆, f(0) =g(0),andf(∆) ⊂g(∆),then we see that the functionf(z)is subordinate tog(z)in∆,and we writef(z)≺g(z).

Definition 1.2 (Subordinating Factor Sequence). A sequence{b_n}^∞_n=1of com- plex numbers is called a subordinating factor sequence if, whenever f(z) is analytic , univalent and convex in∆, we have the subordination given by (1.8)

∞

X

n=2

b_na_nzⁿ ≺f(z) (z ∈∆, a₁ = 1).

Lemma 1.1 ([14]). The sequence{b_n}^∞_n=1is a subordinating factor sequence if and only if

(1.9) Re

( 1 + 2

∞

X

n=1

b_nzⁿ )

>0 (z ∈∆).

Lemma 1.2. If (1.10)

∞

X

n=2

σ_n(a, c;α, β)|a_n| ≤1−α

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where, for convenience,

σ_n(a, c;α, β) := (1 +β)(a)_n+ [1−α−a(1 +β)](a)n−1

(c)n−1

(1.11)

(−1≤α <1; β ≥0, n≥2), thenf(z)∈ L(a, c;α, β).

Proof. It suffices to show that β

−Re

≤1−α.

We have β

−Re

≤(1 +β)

≤

(1 +β)P∞ n=2

a(a+1)n−1−a(a)n−1

(c)n−1

|a_n| |z|ⁿ⁻¹

1−P∞ n=2

(a)n−1

(c)n−1 |an| |z|ⁿ⁻¹

≤

(1 +β)P∞ n=2

(a)n−a(a)n−1

(c)n−1

|a_n|

1−P∞ n=2

(a)n−1

(c)n−1 |an| . The last expression is bounded above by1−αif

∞

X

n=2

(1 +β)(a)_n+ [1−α−a(1 +β)](a)_n−1 (c)n−1

|an| ≤1−α

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and the proof is complete.

LetL^?(a, c;α, β)denote the class of functionsf(z)∈ Awhose coefficients satisfy the condition (1.10). We note thatL^?(a, c;α, β)⊆ L(a, c;α, β).

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2. Main Theorem

Employing the techniques used earlier by Srivastava and Attiya [13], Attiya [1]

and Singh [12], we state and prove the following theorem.

Theorem 2.1. Let the functionf(z)defined by (1.1) be in the classL^?(a, c;α, β) where−1≤α < 1 ; β ≥0;a >0;c >0.Also let Kdenote the familiar class of functionsf(z)∈ Awhich are also univalent and convex in∆. Then

(2.1) σ₂(a, c;α, β)

2[1−α+σ₂(a, c;α, β)](f ∗g)(z)≺g(z) (z ∈∆; g ∈ K), and

(2.2) Re(f(z))>−1−α+σ₂(a, c;α, β)

σ₂(a, c;α, β) , (z ∈∆).

The constant _2[1−α+σ^σ²^(a,c;α,β)

2(a,c;α,β)] is the best estimate.

Proof. Letf(z)∈ L^?(a, c;α, β)and letg(z) = z+P∞

n=2c_nzⁿ∈ K. Then (2.3) σ₂(a, c;α, β)

2[1−α+σ₂(a, c;α, β)](f∗g)(z)

= σ2(a, c;α, β)

2[1−α+σ₂(a, c;α, β)] z+

∞

X

n=2

a_nc_nzⁿ

! .

Thus, by Definition1.2, the assertion of our theorem will hold if the sequence (2.4)

σ₂(a, c;α, β)

2[1−α+σ2(a, c;α, β)]a_n ^∞

n=1

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is a subordinating factor sequence, witha₁ = 1.In view of Lemma1.1, this will be the case if and only if

(2.5) Re (

1 + 2

∞

X

n=1

σ₂(a, c;α, β)

2[1−α+σ₂(a, c;α, β)]a_nzⁿ )

>0 (z ∈∆).

Now Re

(

1 + σ₂(a, c;α, β) 1−α+σ₂(a, c;α, β)

∞

X

n=1

a_nzⁿ )

= Re

1 + σ₂(a, c;α, β) 1−α+σ2(a, c;α, β)z

+ 1

1−α+σ₂(a, c;α, β)

∞

X

n=1

σ₂(a, c;α, β)a_nzⁿ )

≥1−

σ₂(a, c;α, β) 1−α+σ₂(a, c;α, β)r

− 1

1−α+σ₂(a, c;α, β)

∞

X

n=1

σn(a, c;α, β)anrⁿ )

.

Sinceσn(a, c;α, β)is an increasing function ofn(n ≥2) 1−

σ₂(a, c;α, β) 1−α+σ2(a, c;α, β)r

− 1

1−α+σ₂(a, c;α, β)

∞

X

n=1

σn(a, c;α, β)anrⁿ )

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>1− σ₂(a, c;α, β)

1−α+σ₂(a, c;α, β)r− 1−α

1−α+σ₂(a, c;α, β)r (|z|=r)

>0.

Thus (2.5) holds true in∆.This proves the inequality (2.1). The inequality (2.2) follows by taking the convex functiong(z) = _1−z^z =z+P∞

n=2zⁿ in (2.1). To prove the sharpness of the constant _2[1−α+σ^σ²^(a,c;α,β)

2(a,c;α,β)], we consider the function f₀(z)∈ L^?(a, c;α, β)given by

(2.6) f₀(z) = z− 1−α

σ2(a, c;α, β)z² (−1≤α <1; β ≥0).

Thus from (2.1), we have

(2.7) σ₂(a, c;α, β)

2[1−α+σ₂(a, c;α, β)]f₀(z)≺ z 1−z. It can easily verified that

(2.8) min

Re

σ₂(a, c;α, β)

2[1−α+σ₂(a, c;α, β)]f₀(z)

=−1

2 (z ∈∆), This shows that the constant _2[1−α+σ^σ²^(a,c;α,β)

2(a,c;α,β)] is best possible.

Corollary 2.2. Let the functionf(z)defined by (1.1) be in the classL^?(1,1;α, β) and satisfy the condition

(2.9)

∞

X

n=2

[n(1 +β)−(α+β)]|an| ≤1−α

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then

β+ 2−α

2(β+ 3−2α)(f ∗g)(z)≺g(z) (2.10)

(−1≤α <1; β ≥0; z ∈∆; g ∈ K) and

(2.11) Re(f(z))>−β+ 3−2α

β+ 2−α , (z ∈∆).

The constant _{2(β+3−2α)}^β+2−α is the best estimate.

Corollary 2.3. Let the functionf(z)defined by (1.1) be in the classL^?(1,1;α,0) and satisfy the condition

(2.12)

∞

X

n=2

(n−α)|an| ≤1−α, then

(2.13) 2−α

6−4α(f∗g)(z)≺g(z) (z ∈∆; g ∈ K) and

(2.14) Re(f(z))>−3−2α

2−α , (z ∈∆).

The constant _6−4α^2−α is the best estimate.

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Puttingα= 0in Corollary2.3, we obtain

Corollary 2.4 ([12]). Let the function f(z) defined by (1.1) be in the class L^?(1,1; 0,0)and satisfy the condition

(2.15)

∞

X

n=2

n|a_n| ≤1 then

(2.16) 1

3(f∗g)(z)≺g(z) (z ∈∆; g ∈ K) and

(2.17) Re(f(z))>−3

2, (z ∈∆).

The constant1/3 is the best estimate.

Corollary 2.5. Let the functionf(z)defined by (1.1) be in the classL^?(2,1;α, β) and satisfy the condition

(2.18)

∞

X

n=2

n[n(1 +β)−(α+β)]|a_n| ≤1−α, then

β+ 2−α

2β+ 5−3α(f ∗g)(z)≺g(z) (2.19)

(−1≤α <1; β ≥0; z ∈∆; g ∈ K)

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and

(2.20) Re(f(z))>−2β+ 5−3α

2(β+ 2−α), (z ∈∆).

The constant _2β+5−3α^β+2−α is the best estimate.

Corollary 2.6. Let the functionf(z)defined by (1.1) be in the classL^?(2,1;α,0) and satisfy the condition

(2.21)

∞

X

n=2

n(n−α)|a_n| ≤1−α, then

(2.22) 2−α

5−3α(f∗g)(z)≺g(z) (z ∈∆; g ∈ K) and

(2.23) Re(f(z))>− 5−3α

2(2−α), (z ∈∆).

The constant _5−3α^2−α is the best estimate.

Puttingα= 0in Corollary2.6, we obtain

Corollary 2.7. Let the functionf(z)defined by (1.1) be in the classL^?(2,1; 0,0) and satisfy the condition

(2.24)

∞

X

n=2

n²|an| ≤1

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then

(2.25) 2

5(f∗g)(z)≺g(z) (z ∈∆; g ∈ K) and

(2.26) Re(f(z))> −5

4 , (z ∈∆).

The constant2/5is the best estimate.

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[2] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hyperge- ometric functions, SIAM J. Math. Anal., 15(4) (1984), 737–745.

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Math., 57(2) (1992), 165–175.

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[10] F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118(1) (1993), 189–196.

[11] F. RØNNING, On starlike functions associated with parabolic regions, Ann Univ. Mariae Curie-Sklodowska Sect. A, 45 (1991), 117–122.

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