In the present paper, the authors investigate a differential inequality defined by multiplier transformation in the open unit diskE={z :|z|<1}

ON STARLIKENESS AND CONVEXITY OF ANALYTIC FUNCTIONS SATISFYING A DIFFERENTIAL INEQUALITY

SUKHWINDER SINGH, SUSHMA GUPTA, AND SUKHJIT SINGH DEPARTMENT OFAPPLIEDSCIENCES

BABABANDASINGHBAHADURENGINEERINGCOLLEGE

FATEHGARHSAHIB-140407 (PUNJAB) INDIA.

ss_billing@yahoo.co.in DEPARTMENT OFMATHEMATICS

SANTLONGOWALINSTITUTE OFENGINEERING& TECHNOLOGY

LONGOWAL-148106 (PUNJAB) INDIA.

sushmagupta1@yahoo.com sukhjit_d@yahoo.com

Received 03 March, 2008; accepted 10 July, 2008 Communicated by S.S. Dragomir

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

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

2000 Mathematics Subject Classification. 30C45.

1. INTRODUCTION

Let A_p denote the class of functions of the form f(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 writeA₁ =A. A functionf ∈ A_p is said to bep-valent starlike of orderα(0≤α < p)inEif

<

zf⁰(z) f(z)

> α, z ∈E.

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

<

1 + zf⁰⁰(z) f⁰(z)

> α, z ∈E.

064-08

Let K_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)and K(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) Ip(n, λ)f(z) = z^p +

∞

X

k=p+1

k+λ p+λ

n

akz^k, (λ≥0, n∈Z).

The operator Ip(n, λ) has recently been studied by Aghalary et.al. [1]. Earlier, the operator I1(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]. 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}and f ∈ A.

A functionf ∈ A_p is said to be in the classS_n(p, λ, α)for allz inE if 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 thatS₀(1,0, α)andS₁(1,0, α)are the usual classes S^∗(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 ∈ A to belong to the classSn(1,0, α) =Sn(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)

whereαandβ are real numbers andM(α, β, γ, λ, p)is a certain real number given in Section 2, for starlikeness and convexity off ∈ A_p. We obtain sufficient conditions forf ∈ A_p to be a member ofSn(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.

2. MAINRESULT

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 allu₂, v₁ ∈R, withv₁ ≤ −(1 +u²₂)/2and for allz ∈ E. If the functionp, p(z) = 1 +p₁z+p₂z² +· · ·, is analytic inEand ifψ(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 thatβ(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

<

I_p(n+ 1, λ)f(z) Ip(n, λ)f(z)

> γ p i.e.,f(z)∈Sn(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)

Ip(n, λ)f(z) =µ+ (1−µ)r(z), z ∈E.

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)

I_p(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] .

Letu = u1+iu2 andv = v1 +iv2, where u1, u2, v1, v2 are reals withv1 ≤ −^1+u₂ ². 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) . It can be easily verified that φ⁰(u₂) = 0implies thatu₂ = 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 haveψ(r(z), zr⁰(z);z)∈Ωfor allz ∈E, butψ(iu2, v1;z)∈/ Ω, in view of (2.5) and (2.6). Therefore, by Lemma 2.1 and (2.2), we conclude that

<

Ip(n+ 1, λ)f(z) I_p(n, λ)f(z)

> γ p.

3. COROLLARIES

By takingp= 1andλ= 0in Theorem 2.2. We have the following corollary.

Corollary 3.1. Letα ≥0,β ≤1and0≤ γ <1be real numbers such thatβ(1−γ) < ¹₂ and β ≤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 takingp= 1, n= 0andλ = 0in Theorem 2.2. We have the following corollary.

Corollary 3.2. Letα ≥0,β ≤1and0≤ γ <1be real numbers such thatβ(1−γ) < ¹₂ and β ≤α. Iff ∈ Asatisfies the condition

<

zf⁰(z) +αz²f⁰⁰(z) (1−β)f(z) +βzf⁰(z)

> M(α, β, γ,0,1), then

<zf⁰(z) f(z) > γ, 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 Theorem 2.2. We have the following corollary.

Corollary 3.3. Letα≥1and ¹₂ < γ < 1be real numbers. Iff ∈ Asatisfies 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 Theorem 2.2, we have the following result of Ravichandran et. al. [9].

Corollary 3.4. Letα≥0and0≤γ <1be real numbers. Iff ∈ Asatisfies the condition

<zf⁰(z) f(z)

1 +αzf⁰⁰(z) f⁰(z)

> M(α,0, γ,0,1), then

<zf⁰(z) f(z) > γ, 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 whenγ = ^α₂, Corollary 3.4 reduces to the result of Li and Owa [5].

By takingp= 1, n= 0andλ = 1in Theorem 2.2, we have the following corollary.

Corollary 3.5. Letα ≥0,β ≤1and0≤ γ <1be real numbers such thatβ(1−γ) < ¹₂ and β ≤α. Iff ∈ Asatisfies the condition

<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 takingp= 1, n= 1andλ = 0in Theorem 2.2, we have the following corollary.

Corollary 3.6. Letα ≥0,β ≤1and0≤ γ <1be real numbers such thatβ(1−γ) < ¹₂ and β ≤α. Iff ∈ Asatisfies the condition

<

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

> M(α, β, γ,0,1), 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 Theorem 2.2, we have the following corollary.

Corollary 3.7. Letα≥0and0≤γ <1be real numbers. Iff ∈ Asatisfies 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 number M(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).

REFERENCES

[1] R. AGHALARY, R.M. ALI, S.B. JOSHIANDV. RAVICHANDRAN, Inequalities for analytic func- tions defined by certain linear operators, International J. of Math. Sci., to appear.

[2] N.E. CHOANDT.H. KIM, Multiplier transformations and strongly close-to-convex functions, Bull.

Korean Math. Soc., 40 (2003), 399–410.

[3] N.E. CHOANDH.M. SRIVASTAVA, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modelling, 37 (2003), 39–49.

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

366.

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

[6] S.S. MILLER AND P.T. MOCANU, Differential subordinations and inequalities in the complex plane, J. Diff. Eqns., 67 (1987), 199–211.

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

[8] S. OWA, C.Y. SHENANDM. OBRADOVI ´C, Certain subclasses of analytic functions, Tamkang J.

Math., 20 (1989), 105–115.

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