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

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Some Properties for an Integral Operator

Serap Bulut vol. 9, iss. 4, art. 115, 2008

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SOME PROPERTIES FOR AN INTEGRAL OPERATOR DEFINED BY AL-OBOUDI

DIFFERENTIAL OPERATOR

SERAP BULUT

Kocaeli University Civil Aviation College Arslanbey Campus

41285 ˙Izmit-Kocaeli, TURKEY EMail:serap.bulut@kocaeli.edu.tr

Received: 08 July, 2008

Accepted: 12 November, 2008

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

Key words: Analytic functions, Differential operator.

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

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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) =Dδf(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]ⁿa_jz^j, (n∈N0 :=N∪ {0}), withDⁿf(0) = 0.

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)

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

A functionf ∈ Ais said to be convex of order αif 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 orderαin U.

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.

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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 operatorI_n(f₁, . . . , f_m) :A^m→ A

I_n(f₁, . . . , f_m)(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, . . . , fm)(z) =

Z z

0

f₁(t) t

k1

· · ·

f_m(t) t

km

dt

introduced in [5]. More details about I₀(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

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introduced in [1].

(iii) For n = 0, m = 1, k₁ = k ∈ [0,1], k₂ = · · · = k_m = 0 and D⁰f₁(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) If k_i ∈ C for1 ≤ i ≤ m, then we have the integral operatorI_n(f₁, . . . , f_m) studied in [7].

In this paper, we investigate some properties for the operatorsI_n on the classes Sⁿ(δ, α)andMⁿ(δ, β).

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2. Some Properties for I

_n

on the Class S

ⁿ

(δ, α)

Theorem 2.1. Let f_i ∈ Sⁿ(δ, α_i) for 1 ≤ i ≤ m with 0 ≤ α_i < 1, δ ≥ 0 and n∈N0. Also letk_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]ⁿaj,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

· · ·

Dⁿfm(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

lnIn(f1, . . . , fm)⁰(z) =k1[lnDⁿf1(z)−lnz] +· · ·+km[lnDⁿfm(z)−lnz].

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By differentiating the above equality, we get In(f1, . . . , fm)⁰⁰(z)

I_n(f₁, . . . , f_m)⁰(z) =

m

X

i=1

ki

(Dⁿfi(z))⁰ Dⁿf_i(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

k_iRe

z(Dⁿf_i(z))⁰ Dⁿf_i(z)

−

m

X

i=1

k_i+ 1.

Sincef_i ∈ Sⁿ(δ, α_i), we get Re

zI_n(f₁, . . . , f_m)⁰⁰(z) In(f1, . . . , fm)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 letk_i >0,1≤i≤m. If

m

X

i=1

k_i ≤ 1 1−α, thenIn(f1, . . . , fm)∈ K(ρ)withρ= 1 + (α−1)Pm

i=1ki.

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Proof. In Theorem2.1, we considerα₁ =· · ·=α_m =α.

Corollary 2.3. Letf ∈ 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 Corollary2.2, 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 Corollary2.3, we considerk = 1.

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3. Some Properties for I

_n

on the class M

ⁿ

(δ, β)

Theorem 3.1. Letf_i ∈ Mⁿ(δ, β_i)for1≤i≤mwithβ_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 Theorem2.1.

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

Corollary 3.2. Letfi ∈ Mⁿ(δ, β)for1≤i≤mwithβ >1. ThenIn(f1, . . . , fm)∈ N(ρ)withρ= 1 + (β−1)Pm

i=1ki andki >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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References

[1] I.W. ALEXANDER, Functions which map the interior of the unit circle upon simple regions, Ann. of Math., 17 (1915), 12–22.

[2] F.M. AL-OBOUDI, On univalent functions defined by a generalized S˘al˘agean operator, Int. J. Math. Math. Sci., (25-28) 2004, 1429–1436.

[3] D. BREAZ, A convexity property for an integral operator on the classS_p(β), J.

Inequal. Appl., (2008), Art. ID 143869.

[4] D. BREAZ, Certain integral operators on the classesM(β_i)andN(β_i), J. In- equal. Appl., (2008), Art. ID 719354.

[5] D. BREAZANDN. BREAZ, Two integral operators, Studia Univ. Babe¸s-Bolyai Math., 47(3) (2002), 13–19.

[6] D. BREAZ, S. OWAANDN. BREAZ, A new integral univalent operator, Acta Univ. Apulensis Math. Inform., 16 (2008), 11–16.

[7] S. BULUT, Sufficient conditions for univalence of an integral operator defined by Al-Oboudi differential operator, J. Inequal. Appl., (2008), Art. ID 957042.

[8] S.S. MILLER, P.T. MOCANUANDM.O. READE, Starlike integral operators, Pacific J. Math., 79(1) (1978), 157–168.

[9] S. OWA AND H.M. SRIVASTAVA, Some generalized convolution proper- ties associated with certain subclasses of analytic functions, J. Inequal. Pure Appl. Math., 3(3) (2002), Art. 42. [ONLINE: http://jipam.vu.edu.

au/article.php?sid=194].

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[10] G. ¸S. S ˘AL ˘AGEAN, Subclasses of univalent functions, Complex Analysis-Fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), Lecture Notes in Math., vol. 1013, Springer, Berlin, 1983, pp. 362–372.

[11] B.A. URALEGADDI, M.D. GANIGI AND S.M. SARANGI, Univalent func- tions with positive coefficients, Tamkang J. Math., 25(3) (1994), 225–230.