In this paper, we investigate some properties for an integral operator defined by Al-Oboudi differential operator

SOME PROPERTIES FOR AN INTEGRAL OPERATOR DEFINED BY AL-OBOUDI DIFFERENTIAL OPERATOR

SERAP BULUT KOCAELIUNIVERSITY

CIVILAVIATIONCOLLEGE

ARSLANBEYCAMPUS

41285 ˙IZMIT-KOCAELI

TURKEY

serap.bulut@kocaeli.edu.tr

Received 08 July, 2008; accepted 12 November, 2008 Communicated by S.S. Dragomir

ABSTRACT. In this paper, we investigate some properties for an integral operator defined by Al-Oboudi differential operator.

Key words and phrases: Analytic functions, Differential operator.

2000 Mathematics Subject Classification. Primary 30C45.

1. INTRODUCTION

LetAdenote the class of all functions of the form

(1.1) f(z) =z+

∞

X

j=2

a_jz^j

which are analytic in the open unit diskU:={z ∈C:|z|<1}, andS :={f ∈ A :f is univalent inU}.

Forf ∈ A, Al-Oboudi [2] introduced the following operator:

(1.2) D⁰f(z) = f(z),

(1.3) D¹f(z) = (1−δ)f(z) +δzf⁰(z) =D_δf(z), δ ≥0,

(1.4) Dⁿf(z) =Dδ(Dⁿ⁻¹f(z)), (n∈N:={1,2,3, . . .}).

Iff is given by(1.1), then from(1.3)and(1.4)we see that (1.5) Dⁿf(z) =z+

∞

X

j=2

[1 + (j−1)δ]ⁿajz^j, (n∈N⁰ :=N∪ {0}), withDⁿf(0) = 0.

190-08

Whenδ = 1, we get S˘al˘agean’s differential operator [10].

A functionf ∈ Ais said to be starlike of orderαif it satisfies the inequality:

Re

zf⁰(z) f(z)

> α (z ∈U)

for some0≤α <1. We say thatf is in the classS^∗(α)for such functions.

A functionf ∈ Ais said to be convex of orderαif it satisfies the inequality:

Re

zf⁰⁰(z) f⁰(z) + 1

> α (z ∈U)

for some0≤α <1. We say thatf is in the classK(α)if it is convex of orderαinU. We note thatf ∈ K(α)if and only ifzf⁰ ∈ S^∗(α).

In particular, the classes

S^∗(0) :=S^∗ and K(0) :=K are familiar classes of starlike and convex functions inU, respectively.

Now, we introduce two new classesSⁿ(δ, α)andMⁿ(δ, β)as follows:

LetSⁿ(δ, α)denote the class of functionsf ∈ Awhich satisfy the condition Re

z(Dⁿf(z))⁰ Dⁿf(z)

> α (z ∈U) for some0≤α <1,δ ≥0, andn∈N0.

It is clear that

S⁰(δ, α)≡ S^∗(α)≡ Sⁿ(0, α), Sⁿ(0,0)≡ S^∗.

LetMⁿ(δ, β)be the subclass ofA, consisting of the functionsf, which satisfy the inequality Re

z(Dⁿf(z))⁰ Dⁿf(z)

< β (z ∈U) for someβ >1,δ≥0, andn ∈N0.

Also, letN(β)be the subclass ofA, consisting of the functionsf, which satisfy the inequality Re

zf⁰⁰(z) f⁰(z) + 1

< β (z ∈U).

It is obvious that

M⁰(δ, β)≡ M(β)≡ Mⁿ(0, β).

The classesM(β)andN(β)were studied by Uralegaddi et al. in [11], Owa and Srivastava in [9], and Breaz in [4].

Definition 1.1. Let n, m ∈ N⁰ and k_i > 0, 1 ≤ i ≤ m. We define the integral operator I_n(f₁, . . . , f_m) :A^m → A

In(f1, . . . , fm)(z) :=

Z z

0

Dⁿf₁(t) t

k1

· · ·

Dⁿf_m(t) t

km

dt, (z∈U), wheref_i ∈ AandDⁿis the Al-Oboudi differential operator.

Remark 1.

(i) Forn = 0, we have the integral operator I₀(f₁, . . . , f_m)(z) =

Z z

0

f₁(t) t

k1

· · ·

f_m(t) t

km

dt

introduced in [5]. More details aboutI₀(f₁, . . . , f_m)can be found in [3] and [4].

(ii) Forn = 0,m = 1,k₁ = 1, k₂ = · · ·= k_m = 0andD⁰f₁(z) := D⁰f(z) = f(z) ∈ A, we have the integral operator of Alexander

I0(f)(z) :=

Z z

0

f(t) t dt introduced in [1].

(iii) Forn = 0, m = 1, k₁ = k ∈ [0,1], k₂ = · · · = k_m = 0 andD⁰f₁(z) := D⁰f(z) = f(z)∈ S, we have the integral operator

I(f)(z) :=

Z z

0

f(t) t

k

dt studied in [8].

(iv) Ifki ∈ Cfor1 ≤ i ≤ m, then we have the integral operator In(f1, . . . , fm)studied in [7].

In this paper, we investigate some properties for the operatorsInon the classesSⁿ(δ, α)and Mⁿ(δ, β).

2. SOME PROPERTIES FORI_nON THECLASSSⁿ(δ, α)

Theorem 2.1. Letf_i ∈ Sⁿ(δ, α_i)for1 ≤i ≤mwith0≤ α_i < 1, δ ≥0andn ∈ N0. Also let k_i >0,1≤i≤m. If

m

X

i=1

k_i(1−α_i)≤1, thenI_n(f₁, . . . , f_m)∈ K(λ)withλ= 1 +Pm

i=1k_i(α_i−1).

Proof. By(1.5), for1≤i≤m, we have Dⁿf_i(z)

z = 1 +

∞

X

j=2

[1 + (j−1)δ]ⁿa_j,iz^j−1, (n∈N0) and

Dⁿf_i(z) z 6= 0 for allz ∈U.

On the other hand, we obtain

I_n(f₁, . . . , f_m)⁰(z) =

Dⁿf₁(z) z

k1

· · ·

Dⁿf_m(z) z

km

forz ∈U. This equality implies that

lnI_n(f₁, . . . , f_m)⁰(z) = k₁lnDⁿf₁(z)

z +· · ·+k_mlnDⁿf_m(z) z or equivalently

lnI_n(f₁, . . . , f_m)⁰(z) =k₁[lnDⁿf₁(z)−lnz] +· · ·+k_m[lnDⁿf_m(z)−lnz]. By differentiating the above equality, we get

I_n(f₁, . . . , f_m)⁰⁰(z) In(f1, . . . , fm)⁰(z) =

m

X

i=1

k_i

(Dⁿf_i(z))⁰ Dⁿfi(z) − 1

z

.

Thus, we obtain

zI_n(f₁, . . . , f_m)⁰⁰(z) I_n(f₁, . . . , f_m)⁰ + 1 =

m

X

i=1

k_iz(Dⁿf_i(z))⁰ Dⁿf_i(z) −

m

X

i=1

k_i+ 1.

This relation is equivalent to Re

zI_n(f₁, . . . , f_m)⁰⁰(z) I_n(f₁, . . . , f_m)⁰ + 1

=

m

X

i=1

kiRe

z(Dⁿf_i(z))⁰ Dⁿf_i(z)

−

m

X

i=1

ki+ 1.

Sincef_i ∈ Sⁿ(δ, α_i), we get Re

zI_n(f₁, . . . , f_m)⁰⁰(z) I_n(f₁, . . . , f_m)⁰ + 1

>

m

X

i=1

k_iα_i −

m

X

i=1

k_i + 1 = 1 +

m

X

i=1

k_i(α_i−1).

So, the integral operatorI_n(f₁, . . . , f_m)is convex of orderλwithλ= 1 +Pm

i=1k_i(α_i−1).

Corollary 2.2. Letf_i ∈ Sⁿ(δ, α)for1 ≤ i≤ mwith0 ≤α < 1,δ ≥ 0andn ∈N0. Also let k_i >0,1≤i≤m. If

m

X

i=1

ki ≤ 1 1−α, thenI_n(f₁, . . . , f_m)∈ K(ρ)withρ= 1 + (α−1)Pm

i=1k_i.

Proof. In Theorem 2.1, we considerα₁ =· · ·=α_m =α.

Corollary 2.3. Let f ∈ Sⁿ(δ, α) with 0 ≤ α < 1, δ ≥ 0 and n ∈ N0. Also let 0 < k ≤ 1/(1−α). Then the function

I_n(f)(z) = Z z

0

Dⁿf(t) t

k

dt is inK(1 +k(α−1)).

Proof. In Corollary 2.2, we considerm = 1andk₁ =k.

Corollary 2.4. Letf ∈ Sⁿ(δ, α). Then the integral operator I_n(f)(z) =

Z z

0

(Dⁿf(t)/t)dt∈ K(α).

Proof. In Corollary 2.3, we considerk = 1.

3. SOME PROPERTIES FORI_nON THE CLASSMⁿ(δ, β)

Theorem 3.1. Letf_i ∈ Mⁿ(δ, β_i)for1 ≤ i ≤ m withβ_i > 1. ThenI_n(f₁, . . . , f_m) ∈ N(λ) withλ = 1 +Pm

i=1k_i(β_i−1)andk_i >0,(1≤i≤m).

Proof. Proof is similar to the proof of Theorem 2.1.

Remark 2. Forn = 0, we have Theorem2.1in [4].

Corollary 3.2. Letf_i ∈ Mⁿ(δ, β)for1 ≤ i ≤ m withβ > 1. ThenI_n(f₁, . . . , f_m) ∈ N(ρ) withρ= 1 + (β−1)Pm

i=1k_i andk_i >0,(1≤i≤m).

Corollary 3.3. Letf ∈ Mⁿ(δ, β)withβ >1. Then the integral operator I_n(f)(z) =

Z z

0

Dⁿf(t) t

k

dt∈ N(1 +k(β−1)) andk > 0.

Corollary 3.4. Letf ∈ Mⁿ(δ, β)withβ >1. Then the integral operator I_n(f)(z) =

Z z

0

Dⁿf(t)

t dt ∈ N(β).

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