Univalence Conditions for Two General Integral Operators ()

Adriana Oprea, Daniel Breaz

Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, Alba Iulia, Romania.

Department of Mathematics, University of Pite?ti, Pite?ti, Romania.

**DOI: **10.4236/apm.2014.48054
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Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, Alba Iulia, Romania.

Department of Mathematics, University of Pite?ti, Pite?ti, Romania.

Let *A* be the class of all
analytic functions which are analytic in the open unit disc . In this paper we study
the problem of univalence for the following general integral operators:

Share and Cite:

Oprea, A. and Breaz, D. (2014) Univalence Conditions for Two General Integral Operators. *Advances in Pure Mathematics*, **4**, 487-493. doi: 10.4236/apm.2014.48054.

**Keywords:**Analytic Functions, Integral Operators, General Schwarz Lemma

1. Introduction

Let be the unit disk and A be the class of all functions of the form

(1)

which are analytic in U and satisfy the conditions

.

We denote by S the class of univalent and regular functions.

In order to derive our main results, we have to recall here the following univalence conditions.

Theorem 1.1. [1] (Becker’s univalence criterion).

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

, (2)

then the function f is univalent in U.

Theorem 1.2. [2] If the function g is regular in U and in U, then for all the following inequalities hold

(3)

and

the equalities hold in case where and.

Remark 1.3. [2] For, from inequality (3) we obtain for every

(4)

and, hence

(5)

Considering and, then

for all.

2. Main Results

In this paper we study the univalence of the following general integral operators:

(6)

where and,

(7)

where and.

Theorem 2.1. Let, , , , , , If

(8)

for all, for all and

(9)

(10)

where

then the function

(11)

is in the class S.

Proof. We have, , for all and, when.

Let us consider the function:

(12)

From (6), we have:

(13)

and

(14)

From (13) and (14), we have:

Using relations before the function h has the form:

(15)

We have:

By using the relations (15), (8) and (9), we obtain:

(16)

(17)

Applying Remark 1.3 for the function h, we obtain:

(18)

From (18), we get:

(19)

for all.

Let us consider the function:

Since, it results:

Using this result and the form (19), we have:

(20)

for all.

Applying the condition (10) in relation (20), we obtain:

for all and from Theorem 1.1, we have.

Corollary 2.2. Let be a complex number and the functions, , ,.

If

(21)

for all and the constant satisfies the condition:

(22)

then the function

(23)

is in the class S.

Proof. We consider in Theorem 2.1.

Remark 2.3. For, , and in relation (11), we obtain the integral operator

, introduced by J. W. Alexander in [3] .

Remark 2.4. For, , , in relation (6), we obtain the integral operator

, defined and studied by V. Pescar in [4] [5] .

Remark 2.5. For, for all, we get the integral operator, studied by D. Breaz, N. Breaz in [6] and D. Breaz in [7] .

Theorem 2.6.

Let, , , , , ,.

If

(24)

for all, for all and

(25)

(26)

where

then the function

(27)

is in the class S.

Proof. We have, for all and, when.

Let us consider the function:

(28)

From (27), we have:

(29)

and

(30)

From (29) and (30), we get:

(31)

Using relation (31) the function p has the form:

We have:

By using the relations (24), (25) and (28), we obtain:

(32)

and

(33)

Applying Remark 1.3 for the function p, we obtain:

(34)

From (34), we get:

(35)

for all.

Let us consider the function

Since, it results:

Using this result and the form (35), we have:

(36)

for all.

Applying the condition (26) in relation (36), we obtain:

for all and from Theorem 1.1, we have.

Corollary 2.7. Let be a complex number and the functions, , ,.

If

(37)

for all and the constant satisfies the condition:

(38)

then the function

(39)

is in the class S.

Proof. We consider in Theorem 2.6.

Remark 2.8. For, , , in relation (27), we obtain the integral operator

, defined and studied by V. Pescar in [8] [9] .

Remark 2.9. For and in relation (27), we obtain the integral operator

, introduced and studied by N. Ularu and D. Breaz in [10] and [11] .

Acknowledgements

This work was supported by the strategic project PERFORM, POSDRU 159/1.5/S/138963, inside POSDRU Romania 2014, co-financed by the European Social Fund-Investing in People.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | Becker, J. (1972) Lownersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen. Journal für die Reine und Angewandte Mathematik, 255, 23-43. |

[2] | Nehari, Z. (1952) Conformal Mapping. McGraw-Hill Book Company, New York. |

[3] | Alexander, J.W. (1915) Functions Which Map the Interior of the Unit Circle upon Simple Regions. Annals of Mathematics, 17, 12-22. http://dx.doi.org/10.2307/2007212 |

[4] | Pescar, V. (1997) On Some Integral Operations Which Preserve the Univalence. Journal of Mathematics, 30, 1-10. |

[5] | Pescar, V. (1998) On the Univalence of an Integral Operator. Studia Universitatis “Babes-Bolyai”, Cluj-Napoca, Mathematica, 43, 95-97. |

[6] | Breaz, D. and Breaz, N. (2002) Two Integral Operators. Studia Universitatis “Babes-Bolyai”, Cluj-Napoca, Mathematica, 3, 13-21. |

[7] | Breaz, D. (2008) Certain Integral Operators on the Classes M(βi) and N(βi). Journal of Inequalities and Applications, Article ID: 719354. |

[8] | Pescar, V. (1997) Some Integral Operators and Their Univalence. The Journal of Analysis, 5, 157-162. |

[9] | Pescar, V. (1997) An Integral Operator Which Preserves the Univalency. The Annual Conference of the Romanian Society of Mathematical Sciences, Bucharest, 29 May-1 June 1997, 179-181. |

[10] | Ularu, N. and Breaz, D. (2012) Univalence Criterion and Convexity for an Integral Operator. Applied Mathematics Letter, 25, 658-661. http://dx.doi.org/10.1016/j.aml.2011.10.011 |

[11] | Ularu, N. and Breaz, D. (2013) Univalence Condition and Properties for Two Integral Operators. Applied Sciences, 15, 112-117. |

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