(1)SOME APPLICATIONS OF THE GENERALIZED BERNARDI–LIBERA–LIVINGSTON INTEGRAL OPERATOR ON UNIVALENT FUNCTIONS M

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SOME APPLICATIONS OF THE GENERALIZED

BERNARDI–LIBERA–LIVINGSTON INTEGRAL OPERATOR ON UNIVALENT FUNCTIONS

M. ESHAGHI GORDJI, D. ALIMOHAMMADI, AND A. EBADIAN DEPARTMENT OFMATHEMATICS

FACULTY OFSCIENCE

SEMNANUNIVERSITY

SEMNAN, IRAN

madjideshaghi@gmail.com DEPARTMENT OFMATHEMATICS

ARAKUNIVERSITY

ARAK, IRAN

d-alimohammadi@araku.ac.ir DEPARTMENT OFMATHEMATICS

FACULTY OFSCIENCE

URMIAUNIVERSITY

URMIA, IRAN

ebadian.ali@gmail.com

Received 17 October, 2008; accepted 24 July, 2009 Communicated by S.S. Dragomir

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.

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

2000 Mathematics Subject Classification. 30C45, 30C50.

1. INTRODUCTION

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

n=2a_nzⁿwhich are analytic in the unit disk U = {z : |z| < 1}. Also, letS denote the subclass ofA consisting of all univalent functions in U. Supposeλ is a real number with0 ≤ λ < 1. A functionf ∈ S is said to be starlike of orderλif and only ifRen

zf⁰(z) f(z)

o

> λ, z ∈ U. Also,f ∈S is said to be convex of orderλ 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 orderλrespectively. It is well known thatf ∈C(λ)if and only

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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ifzf (λ). If f ∈ A,then f ∈ K(β, λ)if and only if there exists a function g ∈ S (λ)such thatRen_zf0(z)

g(z)

o

> β, z ∈ U,where0 ≤ β < 1. These functions are called close-to-convex functions of orderβ typeλ. A functionf ∈Ais 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 that f ∈ K^∗(β, γ)if and only if zf⁰ ∈ K(β, γ)[9]. For f ∈ A if for some λ(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 by S^∗(η, λ). Iff ∈Asatisfies the condition

(1.2)

arg

1 + zf⁰⁰(z) f⁰(z) −λ

< π

2η, (z ∈U)

for some λand ηas above, then we say thatf(z)is strongly convex of orderη and type λin U and we denote this class by C(η, λ). Clearly f ∈ C(η, λ) if and only ifzf^0∗(η, λ), and in particular, we haveS^∗(1, λ) =S^∗(λ)andC(1, λ) = C(λ).

Forc >−1andf ∈Athe 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 = 1by Libera [4] (see also [8]). The classesST_c(η, λ)andCV_c(η, λ)were introduced by Liu [7], where

ST_c(η, λ) =

f ∈A:L_cf ∈S^∗(η, λ),z(L_cf(z))⁰

L_cf(z) 6=λ, z ∈U

, CVc(η, λ) =

f ∈A:Lcf ∈C(η, λ),(z(L_cf(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)}, Cc(λ) ={f ∈A :Lcf ∈C(λ)}.

Obviously f ∈ CVc(η, λ)if and only if zf⁰ ∈ STc(η, λ). J. L. Liu [5] and [6] introduced and similarly investigated the classesS_σ^∗(λ), C_σ(λ), K_σ(β, λ), K_σ^∗(β, λ), ST_σ(η, λ), CV_σ(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 operatorI^σ was introduced by Jung, Kim and Srivastava [2] and then investigated by Urale- gaddi and Somanatha [13], Li [3] and Liu [5]. For the integral operators 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

anzⁿ,

(1.6) L_cf(z) =z+

∞

X

n=2

c+ 1 n+ca_nzⁿ,

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(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 operator I^σ for any real numberσ. In this paper we investigate the properties of the classes S_c^∗(λ), C_c(λ), K_c(β, λ), K_c^∗(β, λ), ST_c(η, λ)and CV_c(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 function ψ :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=2cnzⁿbe analytic inU so that(p(z), zp⁰(z))∈Dfor allz ∈U. If Re{ψ(p(z), zp⁰(z))}>0, z∈U thenRe{p(z)}>0, z ∈U.

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

n=1c_nzⁿ be analytic in U andp(z) 6= 0, z ∈U.If there exists a pointz₀ ∈U such that|arg(p(z))|< ^π₂ηfor|z|<|z₀|andargp(z₀)|=

π

2η where 0 < η ≤ 1, then ^z⁰_p(z^p⁰^(z⁰⁾

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

2. MAINRESULTS

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

Theorem 2.1.

(i) For f ∈ AifRen

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 function, then S_c^∗(λ)⊂S_c+1^∗ (λ).

(ii) Letc > −λ. Forf ∈AifRe

nzf⁰(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

L_c+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, thenψ(u, v)is a continuous function inD=

C− ^λ+c+1_λ−1 ×Cand(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 haveRe{ψ(p(z), zp⁰(z))} > 0, Lemma 1.1 implies that Rep(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 replacingc+ 1withcwe get the desired result.

Theorem 2.2.

(i) For f ∈ AifRen

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 function, then C_c(λ)⊂C_c+1(λ).

(ii) Letc > −λ. Forf ∈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 Theorem 2.1 we 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)

⇔Lc+1zf^0∗(λ)⇔z(Lc+1f)^0∗(λ)⇔Lc+1f ∈C(λ)⇔f ∈Cc+1(λ).

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

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

Proof. By differentiating logarithmically both sides of(1.3)with respect tozwe obtain

(2.5) z(L_cf(z))⁰

Lcf(z) +c= (c+ 1)f(z) Lcf(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 considerψ(u, v) = u+ _u+c+λ^v . Thenψ is a continuous function inD ={C−(−c−λ)} ×C,(1,0)∈D andRe ψ(1,0)>0. If(iu₂, v₁)∈ Dwith v₁ ≤ −^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 thatRe{p(z)}>0and this completes the proof.

Corollary 2.4. Ifc≥λand ^zf_f(z)⁰^(z) is an analytic function, thenf ∈C(λ)impliesf ∈C_c(λ).

Proof. We have

f ∈C(λ)⇔zf^0∗(λ)Λzf_c^0∗(λ)⇔Lczf⁰ ∈S^∗(λ)

⇔z(L_cf)^0∗(λ)⇔L_cf ∈C(λ)⇔f ∈C_c(λ).

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