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Sufficient Conditions for Univalence of an Integral Operator Defined by Al-Oboudi Differential Operator

Abstract

We investigate the univalence of an integral operator defined by Al-Oboudi differential operator.

1. Introduction

Let denote the class of all functions of the form

(1.1)

which are analytic in the open unit disk , and .

For , Al-Oboudi [1] introduced the following operator:

(1.2)
(1.3)
(1.4)

If is given by (1.1), then from (1.3) and (1.4) we see that

(1.5)

with .

When , we get Sălăgean's differential operator [2].

By using the Al-Oboudi differential operator, we introduce the following integral operator.

Definition 1.1.

Let and , . We define the integral operator ,

(1.6)

where and is the Al-Oboudi differential operator.

Remark 1.2.

  1. (i)

    For , , , , and , we have Alexander integral operator

    (1.7)

which was introduced in [3].

  1. (ii)

    For , , , , and , we have the integral operator

    (1.8)

that was studied in [4].

  1. (iii)

    For , , , , , we have the integral operator

    (1.9)

which was studied in [5].

  1. (iv)

    For , , , and , we have the integral operator

    (1.10)

which was studied in [6, 7].

2. Main Results

The following lemmas will be required in our investigation.

Lemma 2.1 (see [8]).

If the function is regular in the unit disk , , and

(2.1)

for all , then the function is univalent in .

Lemma 2.2 (Schwarz Lemma 2.2) (see [9, page 166]).

Let the analytic function be regular in and let . If, in , , then

(2.2)

and .

The equality holds if and only if and .

Theorem 2.3.

Let , , and , . If

(2.3)

then defined in Definition 1.1 is univalent in .

Proof. .

Since , , by (1.5), we have

(2.4)

for all .

On the other hand, we obtain

(2.5)

for . This equality implies that

(2.6)

or equivalently

(2.7)

By differentiating the above equality, we get

(2.8)

After some calculus, we obtain

(2.9)

By hypothesis, since , and since we have

(2.10)

So, we obtain

(2.11)

Thus .

Remark 2.4.

For , , , we have [5, Theorem 1].

Corollary 2.5.

Let , and , . If

(2.12)

and , then .

Theorem 2.6.

Let , and , . If

(i),

(ii), and

(iii),

then defined in Definition 1.1 is univalent in .

Proof. .

By (2.9), we get

(2.13)

This inequality implies that

(2.14)

By Schwarz lemma (Lemma 2.2), we have

(2.15)

or

(2.16)

for all .

So, by Lemma 2.1, .

Remark 2.7.

For , , , , , we have [7, Theorem 1].

Corollary 2.8.

Let , and , . If

(i),

(ii), and

(iii),

then .

In [10], similar results are given by using the Ruscheweyh differential operator.

References

  1. Al-Oboudi FM: On univalent functions defined by a generalized Sălăgean operator. International Journal of Mathematics and Mathematical Sciences 2004,2004(27):1429–1436. 10.1155/S0161171204108090

    MATH  MathSciNet  Article  Google Scholar 

  2. Sălăgean GŞ: Subclasses of univalent functions. In Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), Lecture Notes in Mathematics. Volume 1013. Springer, Berlin, Germany; 1983:362–372. 10.1007/BFb0066543

    Google Scholar 

  3. Alexander JW: Functions which map the interior of the unit circle upon simple regions. Annals of Mathematics 1915,17(1):12–22. 10.2307/2007212

    MATH  MathSciNet  Article  Google Scholar 

  4. Miller SS, Mocanu PT, Reade MO: Starlike integral operators. Pacific Journal of Mathematics 1978,79(1):157–168.

    MathSciNet  Article  MATH  Google Scholar 

  5. Breaz D, Breaz N: Two integral operators. Studia Universitatis Babeş-Bolyai, Mathematica 2002,47(3):13–19.

    MATH  MathSciNet  Google Scholar 

  6. Pescar V: On some integral operations which preserve the univalence. The Punjab University. Journal of Mathematics 1997, 30: 1–10.

    MATH  MathSciNet  Google Scholar 

  7. Pescar V, Owa S: Sufficient conditions for univalence of certain integral operators. Indian Journal of Mathematics 2000,42(3):347–351.

    MATH  MathSciNet  Google Scholar 

  8. Becker J: Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen. Journal für die Reine und Angewandte Mathematik 1972, 255: 23–43.

    MATH  Google Scholar 

  9. Nehari Z: Conformal Mapping. Dover, New York, NY, USA; 1975:vii+396.

    Google Scholar 

  10. Oros GI, Oros G, Breaz D: Sufficient conditions for univalence of an integral operator. Journal of Inequalities and Applications 2008, 2008:-7.

    Google Scholar 

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Correspondence to Serap Bulut.

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Bulut, S. Sufficient Conditions for Univalence of an Integral Operator Defined by Al-Oboudi Differential Operator. J Inequal Appl 2008, 957042 (2008). https://doi.org/10.1155/2008/957042

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Keywords

  • Analytic Function
  • Differential Operator
  • Integral Operator
  • Unit Disk
  • Schwarz Lemma