SOME APPLICATIONS OF THE GENERALIZED BERNARDI–LIBERA–LIVINGSTON INTEGRAL OPERATOR ON UNIVALENT FUNCTIONS

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Bernardi-Libera-Livingston Integral Operator M. Eshaghi Gordji, D. Alimohammadi and A. Ebadian

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SOME APPLICATIONS OF THE GENERALIZED BERNARDI–LIBERA–LIVINGSTON INTEGRAL

OPERATOR ON UNIVALENT FUNCTIONS

M. ESHAGHI GORDJI D. ALIMOHAMMADI

Dept, of Mathematics, Faculty of Science Department of Mathematics Semnan University, Semnan, Iran Arak University, Arak, Iran

EMail:madjideshaghi@gmail.com EMail:d-alimohammadi@araku.ac.ir

A. EBADIAN

Dept. of Mathematics, Faculty of Science Urmia University, Urmia, Iran

EMail:ebadian.ali@gmail.com Received: 17 October, 2008

Accepted: 24 July, 2009

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 30C45, 30C50.

Key words: Analytic function, Integral operator, Univalent function.

Abstract: In this paper by making use of the generalized Bernardi–Libera–Livingston integral operator we introduce and study some new subclasses of univalent functions.

Also we investigate the relations between those classes and the classes which are studied by Jin–Lin Liu.

Acknowledgements: The authors would like to thank the referee for a number of valuable suggestions regarding of a previous version of this paper.

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1. Introduction

LetAbe the class of functions of the form,f(z) = z+P∞

n=2a_nzⁿwhich are analytic in the unit diskU ={z :|z|<1}. Also, letSdenote the subclass ofAconsisting of all univalent functions inU. Supposeλis a real number with0≤λ <1. A function f ∈ S is said to be starlike of order λ if and only if Ren_zf0(z)

f(z)

o

> λ, z ∈ U. Also, f ∈ S is said to be convex of order λ if and only ifRen

1 + ^zf_f0⁰⁰(z)^(z)

o

> λ, z ∈ U. We denote by S^∗(λ), C(λ)the classes of starlike and convex functions of order λ respectively. It is well known that f ∈ C(λ) if and only if zf^0∗(λ). If f ∈ A,thenf ∈K(β, λ)if and only if there exists a functiong ∈ S^∗(λ)such that Re

nzf⁰(z) g(z)

o

> β, z ∈ U, where0 ≤ β < 1. These functions are called close-to- convex functions of order β type λ. A function f ∈ A is called quasi-convex of orderβ typeλif there exists a functiong ∈ C(λ)such thatRen

(zf⁰(z))⁰ g⁰(z)

o

> β. We denote this class byK^∗(β, λ)[10]. It is easy to see thatf ∈K^∗(β, γ)if and only if zf⁰ ∈K(β, γ)[9]. Forf ∈Aif for someλ(0≤λ <1)andη(0< η ≤1)we have (1.1)

arg

zf⁰(z) f(z) −λ

< π

2η, (z ∈U),

thenf(z)is said to be strongly starlike of orderηand typeλinU and we denote this class byS^∗(η, λ). Iff ∈Asatisfies the condition

(1.2)

arg

1 + zf⁰⁰(z) f⁰(z) −λ

< π

2η, (z ∈U)

for someλandηas above, then we say thatf(z)is strongly convex of orderη and typeλinU and we denote this class byC(η, λ). Clearlyf ∈ C(η, λ)if and only if zf^0∗(η, λ), and in particular, we haveS^∗(1, λ) = S^∗(λ)andC(1, λ) = C(λ).

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For c > −1 and f ∈ A the generalized Bernardi–Libera–Livingston integral operatorL_cf is defined as follows

(1.3) L_cf(z) = c+ 1

z^c Z z

0

t^c−1f(t)dt.

This operator forc∈ N = {1,2,3, . . .}was studied by Bernardi [1] and forc = 1 by Libera [4] (see also [8]). The classesST_c(η, λ)and CV_c(η, λ) were introduced by Liu [7], where

ST_c(η, λ) =

f ∈A:L_cf ∈S^∗(η, λ),z(Lcf(z))⁰

L_cf(z) 6=λ, z ∈U

, CV_c(η, λ) =

f ∈A:L_cf ∈C(η, λ),(z(Lcf(z))⁰)⁰

(L_cf(z))⁰ 6=λ, z ∈U

.

Now by making use of the operator given by (1.3) we introduce the following classes.

S_c^∗(λ) ={f ∈A:L_cf ∈S^∗(λ)}, C_c(λ) ={f ∈A:L_cf ∈C(λ)}.

Obviously f ∈ CV_c(η, λ) if and only if zf⁰ ∈ ST_c(η, λ). J. L. Liu [5] and [6]

introduced and similarly investigated the classesS_σ^∗(λ), Cσ(λ),Kσ(β, λ),K_σ^∗(β, λ), STσ(η, λ), CVσ(η, λ)by making use of the integral operatorI^σf given by

(1.4) I^σf(z) = 2^σ zΓ(σ)

Z z

0

log z

t σ−1

f(t)dt, σ >0, f ∈A.

The operator I^σ was introduced by Jung, Kim and Srivastava [2] and then investigated by Uralegaddi and Somanatha [13], Li [3] and Liu [5]. For the integral opera-

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tors given by(1.3)and(1.4)we have verified following relationships.

(1.5) I^σf(z) = z+

∞

X

n=2

2 n+ 1

σ

a_nzⁿ,

(1.6) Lcf(z) =z+

∞

X

n=2

c+ 1 n+canzⁿ, (1.7) z(I^σL_cf(z))^0σf(z)−cI^σL_cf(z),

(1.8) z(L_cI^σf(z))^0σf(z)−cL_cI^σf(z).

It follows from(1.5)that one can define the operatorI^σfor any real numberσ. In this paper we investigate the properties of the classesS_c^∗(λ), C_c(λ), K_c(β, λ), K_c^∗(β, λ), STc(η, λ) andCVc(η, λ). We also study the relations between these classes by the classes which are introduced by Liu in [5] and [6]. For our purposes we need the following lemmas.

Lemma 1.1 ([9]). Let u = u₁ +iu₂, v = v₁ +iv₂ and let ψ(u, v) be a complex functionψ :D⊂C×C→C. Suppose thatψ satisfies the following conditions

(i) ψ(u, v)is continuous inD;

(ii) (1,0)∈DandRe{ψ(1,0)}>0;

(iii) Re{ψ(iu₂, v₁)} ≤0for all(iu₂, v₁)∈Dwithv₁ ≤ −^1+u₂ ²². Letp(z) = 1 +P∞

n=2c_nzⁿbe analytic inU so that(p(z), zp⁰(z)) ∈ Dfor all z ∈U. IfRe{ψ(p(z), zp⁰(z))}>0, z∈U thenRe{p(z)}>0, z ∈U.

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Lemma 1.2 ([11]). Let the function p(z) = 1 + P∞

n=1c_nzⁿ be analytic inU and p(z) 6= 0, z ∈ U. If there exists a point z₀ ∈ U such that|arg(p(z))| < ^π₂η for

|z| < |z₀| and argp(z₀)| = ^π₂η where 0 < η ≤ 1, then ^z⁰_p(z^p⁰^(z⁰⁾

0) = ikη and k ≥

1

2(r+ ¹_r)when argp(z₀) = ^π₂η,Also, k ≤ ⁻¹₂ (r+ ¹_r)whenargp(z₀) = ^−π₂ η, and p(z₀)^1/η =±ir(r >0).

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2. Main Results

In this section we obtain some inclusion theorems by following the method of proof adopted in [12].

Theorem 2.1.

(i) Forf ∈ Aif Ren

zf⁰(z)

f(z) − ^z(L_L^c^f(z))⁰

cf(z)

o

> 0and ^z(L_L^c+1^f^(z))⁰

c+1f(z) is an analytic func- tion, thenS_c^∗(λ)⊂S_c+1^∗ (λ).

(ii) Letc > −λ. For f ∈ AifRen

zf⁰(z)

f(z) − ^z(L_L^c+1^f(z))⁰

c+1f(z)

o

>0and ^z(L_L^c+1^f^(z))⁰

c+1f(z) is an analytic function, thenS_c+1^∗ (λ)⊂S_c^∗(λ).

Proof. (i) Suppose thatf ∈S_c^∗(λ)and set

(2.1) z(L_c+1f(z))⁰

Lc+1f(z) −λ= (1−λ)p(z), wherep(z) = 1 +P∞

n=2c_nzⁿ. An easy calculation shows that (2.2)

z(Lc+1f(z))⁰ Lc+1f(z)

h

2 +c+^z(L_(L^c+1^f^(z))⁰⁰

c+1f(z))⁰

i

z(Lc+1f(z))⁰

Lc+1f(z) +c+ 1 = zf⁰(z) f(z) . By settingH(z) = ^z(L_L^c+1^f^(z))⁰

c+1f(z) we have (2.3) 1 + z(L_c+1f(z))⁰⁰

(L_c+1f(z))⁰ =H(z) + zH⁰(z) H(z) .

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By making use of(2.3)in(2.2),sinceH(z) =λ+ (1−λ)p(z), we obtain (2.4) (1−λ)p(z) + (1−λ)zp⁰(z)

λ+c+ 1 + (1−λ)p(z) = zf⁰(z) f(z) −λ.

If we consider

ψ(u, v) = (1−λ)u+ (1−λ)v

λ+c+ 1 + (1−λ)u, then ψ(u, v) is a continuous function in D =

C−^λ+c+1_λ−1 ×C and(1,0) ∈ D.

Also,ψ(1,0)>0and for all(iu₂, v₁)∈Dwithv₁ ≤ −^1+u₂ ²² we have Reψ(iu₂, v₁) = (1−λ)(λ+c+ 1)v₁

(1−λ)²u²₂+ (λ+c+ 1)²

≤ −(1−λ)(λ+c+ 1)(1 +u²₂) 2[(1−λ)²u²₂+ (λ+c+ 1)²] <0.

Therefore the function ψ(u, v) satisfies the conditions of Lemma 1.1 and since in view of the assumption, by considering (2.4),we have Re{ψ(p(z), zp⁰(z))} > 0, Lemma1.1implies thatRep(z)>0, z ∈U and this completes the proof of (i).

(ii) For proving this part of the theorem, we use the same method and a easily verified formula similar to (2.2). By replacing c+ 1 with c we get the desired result.

Theorem 2.2.

(i) Forf ∈ Aif Ren

zf⁰(z)

f(z) − ^z(L_L^c^f(z))⁰

cf(z)

o

> 0and ^z(L_L^c+1^f^(z))⁰

c+1f(z) is an analytic func- tion, thenCc(λ)⊂Cc+1(λ).

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(ii) Letc > −λ. For f ∈ AifRen

zf⁰(z)

f(z) − ^z(L_L^c+1^f(z))⁰

c+1f(z)

o

>0and ^z(L_L^c+1^f^(z))⁰

c+1f(z) is an analytic function, thenC_c+1(λ)⊂C_c(λ).

Proof. (i) In view of part (i) of Theorem2.1we can write

f ∈C_c(λ)⇔L_cf ∈C(λ)⇔z(L_cf)^0∗(λ)⇔L_czf^0∗(λ)⇔zf_c^0∗(λ)⇒zf_c+1^0∗ (λ)

⇔L_c+1zf^0∗(λ)⇔z(L_c+1f)^0∗(λ)⇔L_c+1f ∈C(λ)⇔f ∈C_c+1(λ).

Part (ii) of the theorem can be proved in a similar manner.

Theorem 2.3. Ifc≥ −λ and ^zf_f_(z)⁰^(z) is an analytic function, thenf ∈ S^∗(λ)implies f ∈S_c^∗(λ).

Proof. By differentiating logarithmically both sides of (1.3) with respect to z we obtain

(2.5) z(L_cf(z))⁰

L_cf(z) +c= (c+ 1)f(z) L_cf(z) . Again differentiating logarithmically both sides of(2.5)we have

(2.6) p(z) + zp⁰(z)

c+λ+p(z) = zf⁰(z) f(z) −λ, where p(z) = ^z(L_L^c^f(z))⁰

cf(z) −λ. Let us consider ψ(u, v) = u+ _u+c+λ^v . Then ψ is a continuous function inD={C−(−c−λ)} ×C,(1,0)∈D andRe ψ(1,0)>0.

If(iu₂, v₁)∈Dwithv₁ ≤ −^1+u₂ ²²,then

Reψ(iu₂, v₁) = v₁(c+λ)

u²₂+ (c+λ)² ≤0.

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Sincef ∈S^∗(λ),then(2.6)gives

Re(ψ(p(z), zp⁰(z))) = Re

zf⁰(z) f(z) −λ

>0.

Therefore Lemma 1.1 concludes that Re{p(z)} > 0 and this completes the proof.

Corollary 2.4. Ifc ≥ λand ^zf_f_(z)⁰^(z) is an analytic function, then f ∈ C(λ)implies f ∈C_c(λ).

Proof. We have

f ∈C(λ)⇔zf^0∗(λ)Λzf_c^0∗(λ)⇔L_czf⁰ ∈S^∗(λ)

⇔z(L_cf)^0∗(λ)⇔L_cf ∈C(λ)⇔f ∈C_c(λ).

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References

[1] S.D. BERNARDI, Convex and starlike univalent functions, Trans. Amer. Math.

Soc., 135 (1969), 429–446.

[2] I.B. JUNG, Y.C. KIMANDH.M. SRIVASTAVA, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl., 176 (1993), 138–147.

[3] J.L. LI, Some properties of two integral operators, Soochow. J. Math., 25 (1999), 91–96.

[4] R.J. LIBERA, Some classes of regular functions, Proc. Amer. Math. Soc.,16 (1965), 755–758.

[5] J.L. LIU, A linear operator and strongly starlike functions, J. Math. Soc. Japan, 54(4) (2002), 975–981.

[6] J.L. LIU, Some applications of certain integral operator, Kyungpook Math. J., 43(2003), 21–219.

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Math. Sci., 30(9) (2002), 569–574.

[8] A.E. LIVINGSTON, On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc., 17 (1996), 352–357.

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[10] K.I. NOOR, On quasi-convex functions and related topics, Internat. J. Math.

Math. Sci.,10 (1987), 241–258.

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[11] M. NUNOKAWA, S. OWA, H. SAITOH, A. IKEDA AND N. KOIKE, Some results for strongly starlike functions, J. Math. Anal. Appl., 212 (1997), 98–

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[13] B.A. URALEGADDI AND C. SOMANATHA, Certain integral operators for starlike functions, J. Math. Res. Expo., 15 (1995), 14–16.