Vol.4 No.8(2014), Article ID:49143,7 pages DOI:10.4236/apm.2014.48054

Univalence Conditions for Two General Integral Operators

1Department of Mathematics, University of Pitești, Pitești, Romania

2Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, Alba Iulia, Romania

Received 5 June 2014; revised 1 July 2014; accepted 10 July 2014

ABSTRACT

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:

in the open unit disc U, when, , ,.

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.

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