A new univalent integral operator defined by Al-Oboudi differential operator

A new univalent integral operator defined by Al-Oboudi differential operator

Serap Bulut

Abstract

In [3], Breaz and Breaz gave an univalence condition of the integral operator Gn,α introduced in [2]. The purpose of this paper is to give univalence condition of the generalized integral operatorGn,m,αdefined in [4]. Our results generalize the results of [3].

2000 Mathematics Subject Classification: 30C45.

Key words and phrases:Analytic functions, Univalent functions, Integral operator, Differential operator, Schwarz lemma.

1 Introduction

LetA denote the class of all functions of the form

(1) f(z) =z+

X∞

k=2

a_kz^k

1Received 20 November, 2008

Accepted for publication (in revised form) 3 December, 2008

85

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

Forf ∈ A, Al-Oboudi [1] introduced the following operator:

(2) D⁰f(z) =f(z),

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

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

Iff is given by (1), then from (3) and (4) we see that (5) Dⁿf(z) =z+

X∞

k=2

[1 + (k−1)δ]ⁿa_kz^k, (n∈N₀ :=N∪ {0}),

withDⁿf(0) = 0.

Remark 1 When δ= 1, we get S˘al˘agean’s differential operator [9].

The following results will be required in our investigation.

General Schwarz Lemma. [5] Let the function f be regular in the disk U_R={z∈C:|z|< R},with |f(z)|< M for fixed M. If f has one zero with multiplicity order bigger than m for z= 0, then

|f(z)| ≤ M

R^m |z|^m (z∈U_R).

The equality can hold only if

f(z) =e^iθ M R^mz^m, where θ is constant.

Theorem A. [7] Let α be a complex number with Reα >0 and f ∈ A. If f satisfies

1− |z|^2Reα Reα

¯¯

¯¯zf⁰⁰(z) f⁰(z)

¯¯

¯¯≤1 (z∈U),

then, for any complex number β with Reβ ≥Reα, the integral operator

F_β(z) =

½ β

Z _z

0

t^β−1f⁰(t)dt

¾¹

β

is in the class S.

Theorem B.[6] Let f ∈ Asatisfy the following inequality:

(6)

¯¯

¯¯z²f⁰(z) (f(z))² −1

¯¯

¯¯≤1 (z∈U).

Then f is univalent in U.

Theorem C. [8] Assume that g ∈ A satisfies condition (6), and let α be a complex number with

|α−1| ≤ Reα 3 . If

|g(z)| ≤1, ∀z∈U then the function

(7) G_α(z) =

½ α

Z _z

0

(g(t))^α−1dt

¾¹

α

is of class S.

In [2], Breaz and Breaz considered the integral operator (8) G_n,α(z) :=

½

[n(α−1) + 1]

Z _z

0

(g₁(t))^α−1· · ·(g_n(t))^α−1dt

¾ ¹

n(α−1)+1

,

(g₁, . . . , g_n∈ A), and proved that the functionG_n,α is univalent inU.

Remark 2 Note that for n = 1, we obtain the integral operator G_α defined by (7).

Theorem D. [3] Let g_i ∈ A, ∀i= 1, . . . , n, n∈N, satisfy the properties

¯¯

¯¯z²g_i⁰(z) (g_i(z))² −1

¯¯

¯¯<1, ∀z∈U, ∀i= 1, . . . , n

and α∈C with

|α−1| ≤ Reα 3n . If

|g_i(z)| ≤1, ∀z∈U, ∀i= 1, . . . , n, then the function G_n,α defined by (8) is univalent.

In [4], the author introduced a new general integral operator by means of the Al-Oboudi differential operator as follows.

Definition 1 [4] Let n ∈ N, m ∈ N₀ and α ∈ C. We define the integral operator G_n,m,α by

(9)

G_n,m,α(z) :=



[n(α−1) + 1]

Z _z

0

Yn

j=1

(D^mg_j(t))^α−1dt





1 n(α−1)+1

(z∈U),

where g₁, . . . , g_n∈ A and D^m is the Al-Oboudi differential operator.

Remark 3 In the special case n= 1, we obtain the integral operator

(10) G_m,α(z) :=

½ α

Z _z

0

(D^mg(t))^α−1dt

¾¹

α (z∈U).

Remark 4 If we set m = 0 in (9) and (10), then we obtain the integral operators defined in(8) and (7), respectively.

2 Main Results

Theorem 1 Let M_j ≥ 1, each of the functions g_j ∈ A (j ∈ {1, . . . , n}) satisfies the inequality

(11)

¯¯

¯¯

¯

z²(D^mg_j(z))⁰ (D^mg_j(z))² −1

¯¯

¯¯

¯≤1 (z∈U; m∈N₀).

and α∈C with

|α−1| ≤ Reα P_n

j=1(2M_j+ 1), Re(n(α−1) + 1)≥Reα >0.

If

|D^mg_j(z)| ≤M_j (z∈U; j∈ {1, . . . , n}),

then the integral operator G_n,m,α defined by (9) is in the univalent function class S.

Proof. Since g_j ∈ A (j∈ {1, . . . , n}), by (5), we have D^mg_j(z)

z = 1 +

X∞

k=2

[1 + (k−1)δ]^ma_k,jz^k−1 (m∈N₀) and

D^mg_j(z) z 6= 0 for all z∈U.

Also we note that

G_n,m,α(z) =



[n(α−1) + 1]

Z _z

0

t^n(α−1) Yn

j=1

µD^mg_j(t) t

¶_α−1 dt





1 n(α−1)+1

.

Define a function

f(z) = Z _z

0

Yn

j=1

µD^mg_j(t) t

¶_α−1 dt.

Then we obtain

(12) f⁰(z) =

Yn

j=1

µD^mg_j(z) z

¶_α−1 . It is clear that f(0) =f⁰(0)−1 = 0.

The equality (12) implies that lnf⁰(z) = (α−1)

Xn

j=1

lnD^mg_j(z) z

or equivalently

lnf⁰(z) = (α−1) Xn

j=1

(lnD^mg_j(z)−lnz). By differentiating above equality, we get

f⁰⁰(z)

f⁰(z) = (α−1) Xn

j=1

µ(D^mg_j(z))⁰ D^mg_j(z) −1

z

¶ .

Hence we obtain

zf⁰⁰(z)

f⁰(z) = (α−1) Xn

j=1

µz(D^mg_j(z))⁰ D^mg_j(z) −1

¶ ,

which readily shows that 1− |z|^2Reα

Reα

¯¯

¯¯zf⁰⁰(z) f⁰(z)

¯¯

¯¯ ≤ 1− |z|^2Reα

Reα |α−1|

Xn

j=1

µ¯¯

¯¯z(D^mg_j(z))⁰ D^mg_j(z)

¯¯

¯¯+ 1

¶

≤ |α−1|

Reα Xn

j=1

ï¯

¯¯

¯

z²(D^mg_j(z))⁰ (D^mg_j(z))²

¯¯

¯¯

¯

¯¯

¯¯D^mg_j(z) z

¯¯

¯¯+ 1

! .

From the hypothesis, we have |g_j(z)| ≤M_j (j ∈ {1, . . . , n} ;z ∈U), then by the General Schwarz Lemma, we obtain that

|g_j(z)| ≤M_j|z| (j∈ {1, . . . , n} ;z∈U).

Then we find 1− |z|^2Reα

Reα

¯¯

¯¯zf⁰⁰(z) f⁰(z)

¯¯

¯¯ ≤ |α−1|

Reα Xn

j=1

ï¯

¯¯

¯

z²(D^mg_j(z))⁰ (D^mg_j(z))² −1

¯¯

¯¯

¯M_j+M_j + 1

!

≤ |α−1|

Reα Xn

j=1

(2M_j+ 1)≤1

since |α−1| ≤ ^Pn ^Reα

j=1(2Mj+1). Applying Theorem A, we obtain that G_n,m,α is in the univalent function class S.

Corollary 1 Let M ≥ 1, each of the functions g_j ∈ A (j ∈ {1, . . . , n}) satisfies the inequality (11) andα ∈C with

|α−1| ≤ Reα

(2M+ 1)n, Re(n(α−1) + 1)≥Reα >0.

If

|D^mg_j(z)| ≤M (z∈U; j ∈ {1, . . . , n}),

then the integral operator G_n,m,α defined by (9) is in the univalent function class S.

Proof. In Theorem 1, we consider M₁=· · ·=M_n=M.

Corollary 2 Let each of the functions g_j ∈ A (j ∈ {1, . . . , n}) satisfies the inequality (11) and α∈C with

|α−1| ≤ Reα

3n , Re(n(α−1) + 1)≥Reα >0.

If

|D^mg_j(z)| ≤1 (z∈U; j∈ {1, . . . , n}),

then the integral operator G_n,m,α defined by (9) is in the univalent function class S.

Proof. In Corollary 1, we consider M = 1.

Remark 5 If we set m= 0 in Corollary 2, then we have Theorem D.

Corollary 3 Let the function g ∈ A satisfies the inequality (11) and α ∈ C with

|α−1| ≤ Reα

3 , Reα >0.

If

|D^mg(z)| ≤1 (z∈U),

then the integral operator G_m,α defined by (10) is in the univalent function class S.

Proof. In Corollary 2, we consider n= 1.

Remark 6 If we set m= 0 in Corollary 3, then we have Theorem C.

References

[1] F. M. Al-Oboudi,On univalent functions defined by a generalized S˘al˘agean operator, Int. J. Math. Math. Sci. 2004, no. 25-28, 1429-1436.

[2] D. Breaz and N. Breaz, Univalence of an integral operator, Mathematica (Cluj)47(2005), no. 70, 35-38.

[3] D. Breaz and N. Breaz, An integral univalent operator, Acta Math. Univ.

Comenian. (N.S.)76(2007), no. 2, 137-142.

[4] S. Bulut, Univalence condition for a new generalization of the family of integral operators, Acta Univ. Apulensis Math. Inform. No.18(2009), 71- 78.

[5] Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.

[6] S. Ozaki and M. Nunokawa,The Schwarzian derivative and univalent func- tions, Proc. Amer. Math. Soc. 33(1972), 392-394.

[7] N. Pascu, An improvement of Becker’s univalence criterion, Proceedings of the Commemorative Session Simion Stoilow, Brasov, (1987), 43-48.

[8] V. Pescar,New criteria for univalence of certain integral operators, Demon- stratio Math. 33(2000), no. 1, 51-54.

[9] 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.

Serap Bulut Kocaeli University Civi Aviation College Arslanbey Campus

41285 ˙Izmit-Kocaeli, Turkey E-mail: [email protected]