(1)ON CLOSE-TO-CONVEX FOR CERTAIN INTEGRAL OPERATORS Aabed Mohammed, Maslina Darus and Daniel Breaz Abstract

ON CLOSE-TO-CONVEX FOR CERTAIN INTEGRAL OPERATORS

Aabed Mohammed, Maslina Darus and Daniel Breaz

Abstract. In this paper, we consider some sufficient conditions for two integral operators to be close-to-convex function defined in the open unit disk.

Keywords: Analytic functions, closed-to-convex, close-to-star, integral operators.

2000 Mathematics Subject Classification: 30C45

1. Introduction

Let U = {z∈C:|z|<1} be the open unit disk and A denotes the class of functions normalized by

f(z) =z+

∞

X

k=2

a_kz^k (1)

which are analytic in the open unit diskU and satisfy the conditionf(0) =f⁰(0)−1 = 0. We also denote by S the subclass of A consisting of functions which are also univalent inU.

A functionf ∈ Ais the convex function of order ρ, 0 ≤ρ <1, if f satisfies the following inequality

Re

zf⁰⁰(z) f⁰(z) + 1

> ρ, z∈ U

and we denote this class by K(ρ).

Similarly, if f ∈ Asatisfies the following inequality Re

zf⁰(z) f(z)

> ρ, z∈ U

for some ρ, 0 ≤ρ < 1, then f is said to be starlike of orderρ and we denote this class by S^∗(ρ). We note that f ∈ K ⇔zf⁰(z)∈ S^∗,z ∈ U. In particular case, the

classes K(0) =K and S^∗ =S^∗ are familiar classes of starlike and convex functions in U.

A functionf ∈ Ais called close-to-convex if there exist a convex functiongsuch that

Ref⁰(z)

g⁰(z) >0, z ∈ U. (2)

Since f ∈ K ⇔zf⁰(z)∈ S^∗,z∈ U, we can replace (2) by the requirement that Rezf⁰(z)

h(z) >0, z ∈ U,

where h is a starlike function on U. Furthermore,f is closed-to-convex if and only if

θ2

Z

θ1

Re

1 +zf⁰⁰(z) f⁰(z)

dθ >−π,

where 0≤θ₁ < θ₂ ≤2π, z =re^iθ and r <1.

LetCdenote the set of normalized close-to-convex functions onU it is clear that K ⊂ S^∗ ⊂ C ⊂ S.

A function f ∈ Ais called close-to-star inU if there exist a starlike g such that Ref(z)

g(z) >0, z∈ U. Also f is close-to-star in U if and only if

θ2

Z

θ1

Rezf⁰(z)

f(z) dθ >−π,

where 0≤θ1 < θ2 ≤2π, z =re^iθ and r <1. Let CS^∗ denote the class of close-to- star functions in U, it is known that the close-to-star functions are not necessarily univalent inU.

Shukla and Kumar [5] introduced the following subclasses ofC andCS^∗.

A functionf ∈ S belongs to the classC(β, ρ) of close-to-convex functions of order β and type ρ if for someg∈S^∗(ρ),

arg

zf⁰(z) g(z)

< βπ

2 , z∈ U, where β∈[0,1].

A functionf ∈ S belongs to the classCS^∗(β, ρ) of close-to-star functions of order β and type ρ if for someg∈ S^∗(ρ),

arg f(z)

g(z)

< βπ

2 , z ∈ U, where β∈[0,1].

It is clear that C(0, ρ)≡ K(ρ) and CS^∗(0, ρ)≡ S^∗(ρ).

Also C(β, ρ)⊂ C(1,0)≡ C and CS^∗(β, ρ)⊂ CS^∗(1,0)≡ CS^∗. Now, we consider the following integral operators

Fn(z) =

z

Z

0

f1(t) t

α1

·...·

fn(t) t

αn

dt, (3)

and

Fα1,...,αn(z) =

z

Z

0

f₁⁰(t)α1

·...· f_n⁰(t)αn

dt, (4)

where fj ∈ Aand αj >0, for allj ∈ {1,2, ..., n}.

These operators are introduced by D.Breaz and N.Breaz [1] and studied by many authors (see [2], [3], [4]).

In the present paper, we obtain some sufficient conditions for the above integral operators to be in the class of close-to-convex function C.

Before embarking on the proof of our results, we need the following Lemmas introduced by Shukla and Kumar [5].

Lemma 1. If f ∈ S^∗(ρ), then

ρ(θ₂−θ₁)≤

θ2

Z

θ1

Rezf⁰(z)

f(z) dθ≤2π(1−ρ) +ρ(θ₂−θ₁) where z=re^iθ and 0≤θ₁ ≤θ₂≤2π.

Lemma 2. If f ∈ C(β, ρ) then

−βπ+ρ(θ₂−θ₁)≤

θ2

Z

θ1

Re

1 +zf⁰⁰(z) f⁰(z)

dθ≤βπ+ 2π(1−ρ) +ρ(θ₂−θ₁)

where z=re^iθ and 0≤θ₁ ≤θ₂≤2π.

Lemma 3. If f ∈ CS^∗(β, ρ) then

−βπ+ρ(θ2−θ1)≤

θ2

Z

θ1

Rezf⁰(z)

f(z) dθ≤βπ+ 2π(1−ρ) +ρ(θ2−θ1) where z=re^iθ and 0≤θ1 ≤θ2≤2π.

2.Main results

Theorem 1. Let fi ∈ S^∗(ρi), for i∈ {1,2, ..., n}. If

n

P

i=1

αi ≤1, then Fn(z)∈ C, z∈ U, whereFnis defined as in (3), and C is the class of close to convex functions.

Proof. It is clear that F_n⁰(z) 6= 0 for z∈ U. We calculate forFn the derivatives of the first and second order. Since

Fn(z) =

z

Z

0

f1(t) t

α1

·...·

fn(t) t

αn

dt,

then

F_n⁰(z) =

f1(z) z

α1

...

fn(z) z

αn

.

Differentiating the above expression logarithmically, we have F_n⁰⁰(z)

F_n⁰(z) =

n

X

i=1

α_i

f_i⁰(z) f_i(z) −1

z

.

By multiplying the above expression withz we obtain zF_n⁰⁰(z)

F_n⁰(z) =

n

X

i=1

α_i

zf_i⁰(z) fi(z) −1

.

That is equivalent to

1 +zF_n⁰⁰(z) F_n⁰(z) =

n

X

i=1

α_izf_i⁰(z) f_i(z) + 1−

n

X

i=1

α_i. (5)

Taking real parts in (5) and integrating with respect toθ we get

θ2

Z

θ1

Re

1 +zF_n⁰⁰(z) F_n⁰(z)

dθ=

θ2

Z

θ1

n

X

i=1

α_iRe

zf_i⁰(z) f_i(z)

dθ+ 1−

n

X

i=1

α_i

!

(θ₂−θ₁).

Since f_i∈ S^∗(ρ_i) then by applying Lemma 1, we obtain

θ2

Z

θ1

Re

1 +zF_n⁰⁰(z) F_n⁰(z)

dθ≥

n

X

i=1

α_iρ_i−

n

X

i=1

α_i+ 1

!

(θ₂−θ₁).

Since _n

P

i=1

αiρi−

n

P

i=1

αi+ 1

>0 so, minimum is forθ2 =θ1 we obtain that

θ2

Z

θ1

Re

1 +zF_n⁰⁰(z) F_n⁰(z)

dθ >−π,

then F_n(z)∈ C.

Corollary 2. Let fi ∈ S^∗(ρ), for i∈ {1,2, ..., n}. If

n

P

i=1

αi ≤1, then Fn(z) ∈ C, z∈ U, whereF_nis defined as in (3), and C is the class of close to convex functions.

Proof. We consider in Theorem 1, ρ₁ =ρ₂ =...=ρ_n. Theorem 3. Let f_i ∈ CS^∗, i = {1,2, ..., n}. If Pⁿ

i=1

α_i ≤ 1, then F_n(z) ∈ C, z∈ U, whereF_nis defined as in (3), and C is the class of close to convex functions.

Proof. Following the same steps as in Theorem 1, we obtain that

θ2

Z

θ1

Re

1 +zF_n⁰⁰(z) F_n⁰(z)

dθ=

θ2

Z

θ1

n

X

i=1

αiRe

zf_i⁰(z) fi(z)

dθ+ 1−

n

X

i=1

αi

!

(θ2−θ1).

Since f_i∈ CS^∗, then

θ2

Z

θ1

Re

1 +zF_n⁰⁰(z) F_n⁰(z)

dθ >−π

n

X

i=1

αi+ 1−

n

X

i=1

αi

!

(θ2−θ1).

Since 1−Pⁿ

i=1

α_i>0 so, minimum is forθ₁ =θ₂ we obtain that

θ2

Z

θ1

Re

1 +zF_n⁰⁰(z) F_n⁰(z)

dθ >−π

Then Fn∈ C.

Theorem 4. Letf_i∈ C(β_i, ρ_i),i={1,2, ..., n}. If Pⁿ

i=1

α_iβ_i ≤1, thenF_{α1,α2,...,αn}(z)∈ C,z∈ U, whereF_{α1,α2,...,αn} is defined as in (4), andC is the class of close to convex functions.

Proof. It is clear that F_α⁰_1,...,αn(z)6= 0 for z∈ U. Since

Fα1,...,αn(z) =

z

Z

0

f₁⁰(t)α1

·...· f_n⁰(t)αn

dt.

Following the same steps as in Theorem 1, we obtain that zF_α⁰⁰_1,...,αn(z)

F_α⁰_1,...,αn(z) =

n

X

i=1

α_izf_i⁰⁰(z) f_i⁰(z) . That is equivalent to

1 +zF_α⁰⁰_1,...,αn(z) F_α⁰_1,...,αn(z) =

n

X

i=1

αi

zf_i⁰⁰(z) f_i⁰(z) + 1

+ 1−

n

X

i=1

αi. (6)

Taking real parts in (6) and integrating with respect toθ we get

θ2

Z

θ1

Re

1 +zF_α⁰⁰_1,...,αn(z) F_α⁰_1,...,αn(z)

dθ=

θ2

Z

θ1

n

X

i=1

αiRe

zf_i⁰⁰(z) f_i⁰(z) + 1

dθ+ 1−

n

X

i=1

αi

!

(θ2−θ1).

Since f_i∈ C(β_i, ρ_i) then by applying Lemma 2, we obtain

θ2

Z

θ1

Re

1 +zF_α⁰⁰_1,...,αn(z) F_α⁰_1,...,αn(z)

dθ≥

n

X

i=1

αi[−β_iπ+ρi(θ2−θ1)]+ 1−

n

X

i=1

αi

!

(θ2−θ1) =

=

n

X

i=1

αiρi−

n

X

i=1

αi+ 1

!

(θ2−θ1)−π

n

X

i=1

αiβi.

Since

n

P

i=1

α_iρ_i−Pⁿ

i=1

α_i+ 1>0 so, minimum is forθ₁ =θ₂ we obtain that

θ2

Z

θ1

Re

1 +zF_α⁰⁰_1,...,αn(z) F_α⁰_1,...,αn(z)

dθ >−π

then Fα1,...,αn(z)∈ C.

Corollary 5. Letfi ∈ C(β, ρ),i={1,2, ..., n}. If

n

P

i=1

αi≤1, thenFα1,α2,...,αn(z)∈ C,z∈ U, whereFα1,α2,...,αn is defined as in (4), andC is the class of close to convex functions.

Proof. We considet in Theorem 4 β₁ =...=β_n and ρ₁ =...=ρ_n. Theorem 6. If f_i ∈ CS^∗(β_i, ρ_i), i={1,2, ..., n} and

n

P

i=1

α_iβ_i ≤1, then F_n∈ C, where F_n is defined as in (3), and C is the class of close to convex functions.

Proof. Since the proof is similar to the proof of theorems in (1), (3) and (4), it will be omitted.

Corollary 7. If fi ∈ CS^∗(β, ρ), i = {1,2, ..., n} and

n

P

i=1

αi ≤ 1, then Fn ∈ C, where Fn is defined as in (3), and C is the class of close to convex functions.

Proof. We consider in Theorem 6 β₁ =...=β_n and ρ₁ =...=ρ_n.

Acknowledgement: The work here is fully supported by UKM-GUP-TMK- 07-02-107, UKM.

References

[1] D. Breaz and N. Breaz, Two integral operators, Studia Universitatis Babes- Bolyai, Mathematica, 47:3(2002), 13-19.

[2] D. Breaz, S. Owa and N. Breaz, A new integral univalent operator, Acta Universitatis Apulensis, No 16/2008, pp. 11-16.

[3] D. Breaz, A convexity property for an integral operator on the class S_p(β), Journal of Inequalities and Applications, vol. 2008, Article ID 143869.

[4] D. Breaz,Certain Integral Operators On the ClassesM(β_i) andN(β_i), Jour- nal of Inequalities and Applications, vol. 2008, Article ID 719354.

[5] S. L. Shukla and V. Kumar, On The products of close-to-starlike and close- to-convex functions, Indian J. Pure appl. Math. 16(3): (1985), 279-290.

Aabed Mohammed

School of Mathematical Sceinces, Faculty of Science and Technology, Universiti Kebangsaan Malaysia 43600 Bangi, Selangor D. Ehsan, Malaysia

e-mail:[email protected] Maslina Darus

School of Mathematical Sceinces, Faculty of Science and Technology, Universiti Kebangsaan Malaysia 43600 Bangi, Selangor D. Ehsan, Malaysia

e-mail:[email protected] Daniel Breaz

“1 Decembrie 1918” University of Alba Iulia, Faculty of Science,

Department of Mathematics-Informatics, 510009 Alba Iulia,

Romania

e-mail:[email protected]