ON STARLIKENESS AND CONVEXITY OF ANALYTIC FUNCTIONS SATISFYING A DIFFERENTIAL INEQUALITY

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Starlikeness and Convexity of Analytic Functions Sukhwinder Singh, Sushma Gupta

and Sukhjit Singh vol. 9, iss. 3, art. 81, 2008

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ON STARLIKENESS AND CONVEXITY OF ANALYTIC FUNCTIONS SATISFYING A

DIFFERENTIAL INEQUALITY

SUKHWINDER SINGH

Department of Applied Sciences

Baba Banda Singh Bahadur Engineering College Fatehgarh Sahib -140407 (Punjab), INDIA.

EMail:ss_billing@yahoo.co.in

SUSHMA GUPTA AND SUKHJIT SINGH

Department of Mathematics

Sant Longowal Institute of Engineering & Technology Longowal-148106 (Punjab), INDIA.

EMail:sushmagupta1@yahoo.com sukhjit_d@yahoo.com

Received: 03 March, 2008

Accepted: 10 July, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 30C45.

Key words: Multivalent function, Starlike function, Convex function, Multiplier transformation.

Abstract: In the present paper, the authors investigate a differential inequality defined by multiplier transformation in the open unit diskE ={z :|z|<1}. As conse- quences, sufficient conditions for starlikeness and convexity of analytic functions are obtained.

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1. Introduction

LetA_pdenote the class of functions of the formf(z) = z^p+P∞

k=p+1a_kz^k, p∈N= {1,2, . . .}, which are analytic in the open unit discE = {z : |z| < 1}. We write A₁ = A. A functionf ∈ A_p is said to bep-valent starlike of orderα(0 ≤α < p) inE if

<

zf⁰(z) f(z)

> α, z ∈E.

We denote byS_p^∗(α), the class of all such functions. A functionf ∈ A_p is said to be p-valent convex of orderα(0≤α < p)inEif

<

1 + zf⁰⁰(z) f⁰(z)

> α, z ∈E.

LetK_p(α)denote the class of all those functions f ∈ A_p which are multivalently convex of order α in E. Note that S₁^∗(α) and K₁(α) are, respectively, the usual classes of univalent starlike functions of orderα and univalent convex functions of orderα,0≤α <1, and will be denoted here byS^∗(α)andK(α), respectively. We shall useS^∗ andK to denoteS^∗(0)andK(0), respectively which are the classes of univalent starlike (w.r.t. the origin) and univalent convex functions.

Forf ∈ A_p, we define the multiplier transformationI_p(n, λ)as (1.1) I_p(n, λ)f(z) = z^p+

∞

X

k=p+1

k+λ p+λ

n

a_kz^k, (λ≥0, n∈Z).

The operator I_p(n, λ) has recently been studied by Aghalary et.al. [1]. Earlier, the operatorI₁(n, λ)was investigated by Cho and Srivastava [3] and Cho and Kim [2], whereas the operator I₁(n,1)was studied by Uralegaddi and Somanatha [11].

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I₁(n,0)is the well-known S˘al˘agean [10] derivative operatorDⁿ, defined as:Dⁿf(z) = z+P∞

k=2kⁿa_kz^k, n∈N0 =N∪ {0}andf ∈ A.

A functionf ∈ A_p is said to be in the classS_n(p, λ, α)for allzinEif it satisfies

(1.2) <

I_p(n+ 1, λ)f(z) I_p(n, λ)f(z)

> α p,

for someα(0 ≤ α < p, p ∈ N). We note that S₀(1,0, α) and S₁(1,0, α) are the usual classesS^∗(α)andK(α)of starlike functions of orderαand convex functions of orderα, respectively.

In 1989, Owa, Shen and Obradovi˘c [8] obtained a sufficient condition for a functionf ∈ Ato belong to the classS_n(1,0, α) =S_n(α).

Recently, Li and Owa [4] studied the operatorI₁(n,0).

In the present paper, we investigate the differential inequality

<

(1−α)Ip(n+ 1, λ)f(z) +αIp(n+ 2, λ)f(z) (1−β)I_p(n, λ)f(z) +βI_p(n+ 1, λ)f(z)

> M(α, β, γ, λ, p) whereαandβare real numbers andM(α, β, γ, λ, p)is a certain real number given in Section2, for starlikeness and convexity off ∈ A_p. We obtain sufficient conditions forf ∈ A_p to be a member ofS_n(p, λ, γ), for someγ(0 ≤ γ < p, p ∈ N). Many known results for starlikeness appear as corollaries to our main result and some new results regarding convexity of analytic functions are obtained.

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2. Main Result

We shall make use of the following lemma of Miller and Mocanu to prove our result.

Lemma 2.1 ([6,7]). LetΩbe a set in the complex planeCand letψ :C²×E →C. Foru=u₁+iu₂, v =v₁+iv₂, assume thatψsatisfies the conditionψ(iu₂, v₁;z)∈/ Ω, for all u₂, v₁ ∈ R, withv₁ ≤ −(1 +u²₂)/2and for allz ∈ E. If the functionp, p(z) = 1 +p₁z +p₂z² +· · ·, is analytic in E and if ψ(p(z), zp⁰(z);z) ∈ Ω, then

<p(z)>0inE.

We, now, state and prove our main theorem.

Theorem 2.2. Letα ≥ 0,β ≤ 1, λ ≥ 0and0 ≤ γ < pbe real numbers such that β(1− ^γ_p)< ¹₂ andβ ≤α. Iff ∈ A_psatisfies the condition

(2.1) <

(1−α)I_p(n+ 1, λ)f(z) +αI_p(n+ 2, λ)f(z) (1−β)I_p(n, λ)f(z) +βI_p(n+ 1, λ)f(z)

> M(α, β, γ, λ, p), then

<

Ip(n+ 1, λ)f(z) I_p(n, λ)f(z)

> γ p i.e.,f(z)∈S_n(p, λ, γ)where,

M(α, β, γ, λ, p) =

(1−α)γ

p +^αγ_p2² − ^α(¹⁻^γp)

2(p+λ)

1−β

1−^γ_p . Proof. Since0≤γ < p, let us writeµ= ^γ_p. Thus, we have0≤µ <1.

Now we define,

(2.2) I_p(n+ 1, λ)f(z)

I_p(n, λ)f(z) =µ+ (1−µ)r(z), z ∈E.

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Thereforer(z)is analytic inEandr(0) = 1.

Differentiating (2.2) logarithmically, we obtain (2.3) zI_p⁰(n+ 1, λ)f(z)

I_p(n+ 1, λ)f(z) −zI_p⁰(n, λ)f(z)

I_p(n, λ)f(z) = (1−µ)zr⁰(z)

µ+ (1−µ)r(z), z ∈E.

Using the fact that

zI_p⁰(n, λ)f(z) = (p+λ)I_p(n+ 1, λ)f(z)−λI_p(n, λ)f(z).

Thus (2.3) reduces to I_p(n+ 2, λ)f(z)

Ip(n+ 1, λ)f(z) =µ+ (1−µ)r(z) + (1−µ)zr⁰(z) (λ+p)[µ+ (1−µ)r(z)]. Now, a simple calculation yields

(1−α)Ip(n+ 1, λ)f(z) +αIp(n+ 2, λ)f(z) (1−β)I_p(n, λ)f(z) +βI_p(n+ 1, λ)f(z)

=

(1−α) +α

µ+ (1−µ)r(z) + ^(1−µ)zr

0(z) (λ+p)[µ+(1−µ)r(z)]

(1−β) +β[µ+ (1−µ)r(z)] [µ+ (1−µ)r(z)]

=

(1−α)[µ+ (1−µ)r(z)] +α

[µ+ (1−µ)r(z)]²+ ^(1−µ)zr

0(z) (λ+p)

(1−β) +β[µ+ (1−µ)r(z)]

=ψ(r(z), zr⁰(z);z) (2.4)

where,

ψ(u, v;z) =

(1−α)[µ+ (1−µ)u] +α

(µ+ (1−µ)u)²+^(1−µ)v_(λ+p) (1−β) +β[µ+ (1−µ)u] .

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Letu=u₁ +iu₂andv =v₁+iv₂, whereu₁, u₂, v₁, v₂ are reals withv₁ ≤ −^1+u₂ ²². Then, we have

<ψ(iu2, v1;z)

= [(1−α)µ+αµ²][1−β(1−µ)]

[1−β(1−µ)]²+β²(1−µ)²u²₂

+(1−µ)²[(1−α)β−α(1−β(1−µ)) + 2αβµ]u²₂+ α(1−µ)[1−β(1−µ)]v₁ p+λ

[1−β(1−µ)]²+β²(1−µ)²u²₂

≤ h

(1−α)µ+αµ²−^α(1−µ)_2(λ+p)i

[1−β(1−µ)]

[1−β(1−µ)]²+β²(1−µ)²u²₂

+ h

(1−µ)²[(1−α)β−α(1−β(1−µ)) + 2αβµ]−α(1−µ)[1−β(1−µ)]

2(p+λ)

i u²₂

[1−β(1−µ)]² +β²(1−µ)²u²₂

= A+Bu²₂

[1−β(1−µ)]² +β²(1−µ)²u²₂

=φ(u₂), say

≤maxφ(u₂) (2.5)

where,

A=

(1−α)µ+αµ² −α(1−µ) 2(λ+p)

[1−β(1−µ)]

and

B = (1−µ)²[(1−α)β−α(1−β(1−µ)) + 2αβµ]− α(1−µ)[1−β(1−µ)]

2(p+λ) .

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It can be easily verified that φ⁰(u₂) = 0 implies that u₂ = 0. Under the given conditions, we observe thatφ⁰⁰(0)<0. Therefore,

(2.6) maxφ(u₂) =φ(0) =M(α, β, γ, λ, p).

Let

Ω ={w: <w > M(α, β, γ, λ, p)}.

Then from (2.1) and (2.4), we have ψ(r(z), zr⁰(z);z) ∈ Ω for all z ∈ E, but ψ(iu2, v1;z)∈/ Ω, in view of (2.5) and (2.6). Therefore, by Lemma2.1and (2.2), we conclude that

<

I_p(n+ 1, λ)f(z) I_p(n, λ)f(z)

> γ p.

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3. Corollaries

By takingp= 1andλ= 0in Theorem2.2. We have the following corollary.

Corollary 3.1. Let α ≥ 0, β ≤ 1 and 0 ≤ γ < 1 be real numbers such that β(1−γ)< ¹₂ andβ≤α. Iff ∈ Asatisfies the condition

<

(1−α)Dⁿ⁺¹f(z) +αDⁿ⁺²f(z) (1−β)Dⁿf(z) +βDⁿ⁺¹f(z)

> M(α, β, γ,0,1),

then

<Dⁿ⁺¹f(z) Dⁿf(z) > γ, i.e.f(z)∈S_n(γ), where,

M(α, β, γ,0,1) = (1−α)γ+αγ²− ^α(1−γ)₂ 1−β(1−γ) .

By taking p = 1, n = 0 and λ = 0 in Theorem 2.2. We have the following corollary.

<

zf⁰(z) +αz²f⁰⁰(z) (1−β)f(z) +βzf⁰(z)

> M(α, β, γ,0,1),

then

<zf⁰(z) f(z) > γ,

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i.e.f(z)∈S^∗(γ), where,

M(α, β, γ,0,1) = (1−α)γ+αγ²− ^α(1−γ)₂ 1−β(1−γ) .

By takingp= 1, n= 0, λ= 0andβ = 1in Theorem2.2. We have the following corollary.

Corollary 3.3. Let α ≥ 1and ¹₂ < γ < 1be real numbers. Iff ∈ A satisfies the condition

<

1 +αzf⁰⁰(z) f⁰(z)

> M(α,1, γ,0,1), then

<zf⁰(z) f(z) > γ, i.e.f(z)∈S^∗(γ), where

M(α,1, γ,0,1) = 1−α(1−γ)

1 + 1 2γ

By takingp= 1, n = 0, λ= 0andβ = 0in Theorem2.2, we have the following result of Ravichandran et. al. [9].

Corollary 3.4. Letα ≥ 0and 0 ≤ γ < 1be real numbers. Iff ∈ A satisfies the condition

<zf⁰(z) f(z)

1 +αzf⁰⁰(z) f⁰(z)

> M(α,0, γ,0,1), then

<zf⁰(z) f(z) > γ,

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i.e.f(z)∈S^∗(γ), where,

M(α,0, γ,0,1) = (1−α)γ+αγ²−α(1−γ)

2 .

Remark 1. In the case when γ = ^α₂, Corollary 3.4 reduces to the result of Li and Owa [5].

By taking p = 1, n = 0 and λ = 1 in Theorem 2.2, we have the following corollary.

<1 2

(2−α)f(z) + (2 +α)zf⁰(z) +αz²f⁰⁰(z) (2−β)f(z) +βzf⁰(z)

> M(α, β, γ,1,1),

then

<1 2

1 + zf⁰(z) f(z)

> γ,

where,

M(α, β, γ,1,1) = (1−α)γ+αγ²− ^α(1−γ)₄ 1−β(1−γ) .

By taking p = 1, n = 1 and λ = 0 in Theorem 2.2, we have the following corollary.

<

zf⁰(z) + (2α+ 1)z²f⁰⁰(z) +αz³f⁰⁰⁰(z) zf⁰(z) +βz²f⁰⁰(z)

> M(α, β, γ,0,1),

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then

<

1 + zf⁰⁰(z) f⁰(z)

> γ, i.e.f(z)∈K(γ),where,

M(α, β, γ,0,1) = (1−α)γ+αγ²− ^α(1−γ)₂ 1−β(1−γ) .

By takingp= 1, n = 1, λ= 0andβ = 0in Theorem2.2, we have the following corollary.

Corollary 3.7. Letα ≥ 0and 0 ≤ γ < 1be real numbers. Iff ∈ A satisfies the condition

<

1 + (2α+ 1)zf⁰⁰(z)

f⁰(z) +αz²f⁰⁰⁰(z) f⁰(z)

> M(α,0, γ,0,1),

then

<

1 + zf⁰⁰(z) f⁰(z)

> γ,

i.e.,f(z)∈K(γ), where,

M(α,0, γ,0,1) = (1−α)γ+αγ²−α(1−γ)

2 .

Remark 2. In the main result, the real numberM(α, β, γ, λ, p)may not be the best possible as authors have not obtained the extremal function for it. The problem is still open for the best possible real numberM(α, β, γ, λ, p).

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References

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[3] N.E. CHO AND H.M. SRIVASTAVA, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Mod- elling, 37 (2003), 39–49.

[4] J.-L. LIANDS. OWA, Properties of the S˘al˘agean operator, Georgian Math. J., 5(4) (1998), 361–366.

[5] J.-L. LI AND S. OWA, Sufficient conditions for starlikeness, Indian J. Pure Appl. Math., 33 (2002), 313–318.

[6] S.S. MILLERAND P.T. MOCANU, Differential subordinations and inequali- ties in the complex plane, J. Diff. Eqns., 67 (1987), 199–211.

[7] S.S. MILLER AND P.T. MOCANU, Differential Suordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Math- ematics (No. 225), Marcel Dekker, New York and Basel, 2000.

[8] S. OWA, C.Y. SHEN AND M. OBRADOVI ´C, Certain subclasses of analytic functions, Tamkang J. Math., 20 (1989), 105–115.

[9] V. RAVICHANDRAN, C. SELVARAJ AND R. RAJALAKSHMI, Sufficient conditions for starlike functions of orderα, J. Inequal. Pure and Appl. Math.,

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3(5) (2002), Art. 81. [ONLINE:http://jipam.vu.edu.au/article.

php?sid=233].

[10] G.S. S ˘AL ˘AGEAN, Subclasses of univalent functions, Lecture Notes in Math., 1013, 362–372, Springer-Verlag, Heidelberg,1983.

[11] B.A. URALEGADDI AND C. SOMANATHA, Certain classes of univalent functions, in Current Topics in Analytic Function Theory, H.M. Srivastava and S. Owa (ed.), World Scientific, Singapore, (1992), 371–374.