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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 404672, 7 pages
http://dx.doi.org/10.1155/2013/404672
Research Article

A Connection between Basic Univalence Criteria

Department of Mathematics, Faculty of Mathematics and Computer Science, “Transilvania” University of Braşov, 500091 Braşov, Romania

Received 23 January 2013; Accepted 25 June 2013

Academic Editor: Abdelaziz Rhandi

Copyright © 2013 Horiana Tudor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We obtain a sufficient condition for the analyticity and the univalence of a class of functions defined by an integral operator. The well-known univalence criteria of Alexander, Noshiro-Warschawski, Nehari, Goluzin, Ozaki-Nunokawa, Becker, and Lewandowski would follow upon specializing the functions and the parameters involved in the main result. The results obtained not only reduce to those earlier works, but they also extend the previous results.

1. Introduction

Let , be the disk of radius centered at , let be the unit disk, and let .

Denote by the class of analytic functions in which satisfy the usual normalization .

The first results concerning univalence criteria are related to the univalence of an analytic function in the unit disk. Among the most important sufficient conditions for univalence we mention those obtained by Alexander [1], Noshiro [2] and Warschawski [3], Nehari [4], Goluzin [5], Ozaki and Nunokawa [6], Becker [7], and Lewandowski [8].

Furthermore, the first extension of univalence criteria was obtained by Pascu in [9]. In his paper, starting from an analytic function in the unit disk he established not only the univalence of but also the analyticity and the univalence of a whole class of functions defined by an integral operator.

Other extensions of the univalence criteria, for an integral operator, were obtained in the papers [1014]. From the main result of this paper, we found all the univalence criteria mentioned earlier and at the same time other new ones.

2. Loewner Chains

Before proving our main result we need a brief summary of theory of Loewner chains.

A function is said to be a Loewner chain or a subordination chain if(i) is analytic and univalent in for all ;(ii) for all , where the symbol ‘‘’’ stands for subordination.

The following result due to Pommerenke is often used to obtain univalence criteria.

Theorem 1 (see [15, 16]). Let , , be an analytic function in for all , locally absolutely continuous in , locally uniform with respect to . For almost all , suppose that where is analytic in and satisfying for all , . If and forms a normal family in , then for each , the function has an analytic and univalent extension to the whole disk .

3. Main Result

Making use of Theorem 1, the essence of which is the construction of suitable Loewner chain, we can prove our main result.

Theorem 2. Let , and be complex numbers such that , and For , if there exist two analytic functions in such that the inequalities are true for all , then the function , is analytic and univalent in , where the principal branch is intended.

Proof. We consider the function defined by For all and we have , and from the analyticity of in it follows that is also analytic in . Since , there exists a disk , , in which for all . Since , it is easy to see that the function can be written as , where is analytic in , for all , and . It follows that the function is also analytic in a disk , and
Let us prove that for all . We have . From and since , we see that which is equivalent to . It follows that . Assume now that there exists such that . Then . From , , it results that , and from inequality (2), we conclude that for all . Therefore, there is a disk , in which , for all , and we can choose an analytic branch of , denoted by . We fix a determination of , denoted by . For we fix, for , the determination equal to , where From these considerations it follows that the function is analytic in , for all , and can be written as follows: If is the Taylor expansion of in , we have . Since , we have . We saw also that for all .
From the analyticity of in , it follows that there exists a number , and a constant such that and thus is a normal family in . From the analyticity of , for all fixed numbers and , there exists a constant (that depends on and ) such that It follows that the function is locally absolutely continuous in , locally uniform with respect to . The function defined by (1) is analytic in a disk , for all . In order to prove that the function is analytic and has positive real part in , we will show that the function , is analytic in , and Elementary calculation gives From (3) and (4) we deduce that , for all , and then the function is analytic in the unit disk . For , in view of (3), we have In order to evaluate , we will use the following inequality (see [17]): For and , from (15), we have From which is equivalent to and since , we have .
Let be a fixed number, , and let . Since for all , the function is analytic in . Using the maximum modulus principle it follows that for each , arbitrary fixed, there exists such that Denote that . Then , and from (15) we obtain Since , inequality (4) implies that , and from (16), (18), and (19) we conclude that inequality (14) holds true for all and . Since all the conditions of Theorem 1 are satisfied, it follows that is a Loewner chain, for each . For it results that the function is analytic and univalent in , and then the function defined by (5) is analytic and univalent in .

4. Specific Cases and Examples

Suitable choices of the functions and and special values of the parameter yield various types of univalence criteria. So, if in Theorem 2 we take and , we get the following result.

Theorem 3. Let and be complex numbers such that , and . For , if there exists an analytic function in , such that the inequalities are true for all , then the function defined by (5) is analytic and univalent in , where the principal branch is intended.

Theorem 3 gives us a ‘‘continuous’’ passage from Becker’s criterion to Lewandowski’s criterion. Indeed, for , we have the following.

Corollary 4. Let and be complex numbers, , , , and . If for all then the function defined by (5) is analytic and univalent in .

Remark 5. Corollary 4 generalizes the well-known univalence criterion due to Becker. For we found the result from [9]. In the case when and , the previous corollary reduces to Becker’s criterion [7].

For , where is analytic in , , from Theorem 3 we have the following.

Corollary 6. Let and be complex numbers, , , , and . If there exists an analytic function with positive real part in , , such that the inequality is true for all , then the function defined by (5) is analytic and univalent in .

Remark 7. Corollary 6 represents a generalization of the univalence criterion due to Lewandowski. For we found the result from [12]. In the case when and , the previous corollary reduces to Lewandowski’s criterion [8].

For and , from Theorem 2 we can derived some results from paper [18].

Theorem 8. Let and be complex numbers such that , . For , if there exists an analytic function in , , such that the inequalities are true for all , then the function defined by (5) is analytic and univalent in .

Proof. In view of assumption and since , it follows that . But is equivalent to and with . It results that inequality (2) is true. From (3) and (4) we get immediately inequalities (25) and (26).

For and , from Theorem 8 we obtain the following.

Corollary 9. Let be a complex number, . If for all the function satisfies then the function is univalent in . Moreover, it is a spiral-like function.

Proof. For , we have , and in view of (27), inequality (25) of Theorem 8 is verified and inequality (26) is also reduced to (25). It follows that is univalent in . The condition (27) of the corollary can be written as . It follows that . If we put , where from we have , then for all we have , which shows that is spiral-like in .

Taking , we get the following useful corollary which generalizes the result from [19].

Corollary 10. Let and be complex numbers such that , , and . If the inequality holds true for all , then the function defined by (5) is analytic and univalent in .

Proof. It is easy to check that inequality (28) implies inequality (26) of Theorem 8. Indeed, for and making use of (17), we have

For the function , from Theorem 8 we get the following.

Corollary 11. Let and be complex numbers such that , , and . If the inequality holds true for all , then the function defined by (5) is analytic and univalent in . In particular, the function is univalent in , where .

Remark 12. From inequality (30), only for real number, , we get . For complex number, if we put , where from we have , then from inequality (28) we obtain . So, in both cases, we can also conclude that is univalent in from Alexander’s theorem [1], and respectively, from Noshiro-Warschawski’s theorem [2, 3].

Example 1. Consider the function , where , . The condition (30) of Corollary 11 is satisfied. Indeed, since is equivalent with , we get Then, for all complex numbers , , and , by using (5), we obtain that is analytic and univalent in .

If in Theorem 2 we take and , then we have the following.

Theorem 13. Let and be complex numbers such that , , and . For , if there exists an analytic function in , , such that the inequality is true for all , then the function defined by (5) is analytic and univalent in .

For the following results.

Corollary 14. Let and be complex numbers, , , , and . If for all where then the function defined by (5) is analytic and univalent in .

Remark 15. For special values of the parameters and , from Corollary 14 we get some known results. For , we get the result given in [13]. For , since , Corollary 14 generalizes the criterion of univalence due to Nehari, and for and we obtain the univalence criterion due to Nehari [4].

For we have the following.

Corollary 16. Let and be complex numbers, , , , and . If for all then the function defined by (5) is analytic and univalent in .

Remark 17. Corollary 16 represents a generalization of the univalence criterion due to Goluzin. For we obtain the results from paper [11]. For and we get Goluzin’s criterion [5].

For , and , from Theorem 2 we get the following.

Corollary 18. Let and be complex numbers, , , , and . If satisfies the inequalities then the function defined by (5) is analytic and univalent in .

Remark 19. Corollary 18 represents a generalization of the univalence criterion due to Ozaki and Nunokawa. For we found the result from [14]. In the case when and , Corollary 18 reduces to the univalence criterion of Ozaki and Nunokawa [6].

Example 2. Let be a natural number, . We consider the function We note that The condition (37) of Corollary 18 is verified, and it assures the univalence of the function . Taking into account (40), for and , from (38) we have that because the greatest value of the function for , is taken for and is . It follows that all the conditions of Corollary 18 are satisfied, and therefore the function defined by (5) is analytic and univalent in .

Remark 20. Theorem 2 gives us a connection between Alexander’s theorem, Noshiro-Warschawski’s theorem, and the univalence criteria of Becker, Lewandowski, Nehari, Goluzin, and Ozaki and Nunokawa as well as their generalizations.

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