- Research
- Open access
- Published:
p-subordination chains and p-valence criteria
Journal of Inequalities and Applications volume 2013, Article number: 127 (2013)
Abstract
The main object of this investigation is to give some sufficient conditions for analytic functions, by the method of p-subordination chains, to be the p th power of a univalent function in the open unit disk . Also, the significant relationships and relevance to other results are also given. A number of known univalent conditions would follow upon specializing the parameters involved in our main results.
MSC:30C45, 30C55, 30C80.
1 Introduction
Denote by () the disk of radius r, and let . Let denote the class of analytic functions in the open unit disk which satisfy the usual normalization condition . Traditionally, the subclass of consisting of univalent functions is denoted by . Let denote the class of functions , , that satisfy the condition . Also, let denote the class of analytic functions in the open unit disk which satisfy the normalizations for () and , and let be the subclass of consisting of functions of the form in . These classes have been one of the most important subjects of research in geometric function theory for a long time (see [1]). For analytic functions and in , f is said to be subordinate to g, denoted by , if there exists an analytic function w satisfying , , such that (). In particular, if the function g is univalent in , the above subordination is equivalent to and .
2 p-normalized subordination chain and related theorem
Before proving our main theorem, we need a brief summary of the method of p-subordination chains.
Definition 2.1 (see Hallenbeck and Livingston [2])
Let be a function defined on , where . is called a p-subordination chain if satisfies the following conditions:
-
1.
is analytic in for all ,
-
2.
, , and ,
-
3.
for all , .
p-subordination chain is said to be normalized if and for all .
In order to prove our main results, we need the following lemma due to Hallenbeck and Livingston [2].
Lemma 2.1 Let , , be analytic in for all . Suppose that
-
(i)
is a locally absolutely continuous function in the interval ℐ and locally uniform with respect to .
-
(ii)
is a complex-valued continuous function on ℐ such that , for and
forms a normal family of functions in .
-
(iii)
There exists an analytic function satisfying for all , and
(2.1)
Then, for each , the function is the pth power of a univalent function in .
Pommerenke’s theory of subordination chains [3, 4] corresponds to .
The univalence of complex functions is an important property, but, unfortunately, it is difficult and in many cases impossible to show directly that a certain complex function, especially a function belonging to the class , is univalent. Pommerenke [3, 4] and Becker [5] have used the idea of normalized 1-subordination chains, or briefly subordination chains, to obtain sufficient conditions for univalence of the functions belonging to the class . There are three very important criteria for univalence of the function . Two of them are the well-known criteria of Becker [5] and Ahlfors [6] which were obtained by a clever use of the theory of subordination chains and the generalized Loewner differential equation. The other, Nehari’s univalence criterion (see [7]), was obtained without using the subordination chains for the analytic functions. Then Epstein [8] generalized this criterion by using the hyperbolic geometry, and his proof was quite different from the subordination chains method. By using the subordination chains methods, Pommerenke [9] gave a simplified proof of a univalence criterion obtained earlier by Epstein [8]. But in some cases, these criteria may not be sufficient for learning the univalence of the function . For example, although the function is univalent, this function is satisfied neither by Becker and Ahlfors nor by Nehari criteria. This situation is deficiency for these criteria. For this reason, we need to find new criteria or generalize the current criteria. During the time many mathematicians have studied on this problem and have obtained some results (see [10–19] and [20]).
On the other hand, Hallenbeck and Livingston [2] defined p-subordination chains and gave Lemma 2.1 for the functions . In the same paper, they obtained some results for to be the p th power of a univalent function in . Their criteria are a p-valence version of Becker and Ahlfors’s criteria. Recently Deniz et al. [10] submitted a paper which includes sufficient conditions for a integral operator to be the p th power of a univalent function in .
In the present paper, we obtain sufficient conditions for the functions f belonging to the class in terms of the Schwarz derivative defined by
to be the p th power of a univalent function by using p-subordination chains. Our main result is a p-valence version of Nehari [7] and Epstein’s [8] criteria.
3 p-valence criteria
Making use of Lemma 2.1, we can prove now our main result related to the Schwarz derivative.
Theorem 3.1 Let . If
for all , then f is the pth power of a univalent function in .
Proof
Consider the functions defined by
where we choose the branch of the power , which for has value 1, and
The functions u and v are analytic in since f and g analytic.
For all and (), the function defined formally by
is analytic in since is an analytic function in for each fixed and . From (3.4) we have and
After simple calculation, we obtain, for each ,
The limit function belongs to the family ; then there exists a number () such that in every closed disk , there exists a constant such that
uniformly in this disk, provided that t is sufficiently large. Thus, by Montel’s theorem, forms a normal family in each disk .
Since the function is analytic in , for , the function is continuous on the compact set, so , , is a bounded function. Thus, for all fixed , we can write , and we obtain that for all fixed numbers , there exists a constant such that
Therefore, the function is locally absolutely continuous in ℐ; locally uniform with respect to .
After simple calculations, from (3.4) we obtain
and
where
and u, v, , , , are calculated at .
Consider the function for and defined by
From (3.5) to (3.9), we can easily see that the function is analytic in , . If the function
is analytic in and for all and , then has an analytic extension with a positive real part () in for all .
From equality (3.10) we have
where
for and .
The inequality for all and , where is defined by (3.11), is equivalent to
From the hypothesis of theorem, (3.12) and (3.13), we have
and
Since for all and , we find that is an analytic function in . By the maximum modulus principle, it follows that for all and each arbitrarily fixed, there exists such that
Denote . Then , and from (3.7)-(3.9) and (3.12), we have
Because , the inequality (3.1) implies that , and from (3.14), (3.15) and (3.16), we conclude that for all and . Therefore for all and . Since all the conditions of Lemma 2.1 are satisfied, we obtain that the function is the p th power of a univalent function in the whole unit disk for all . □
Theorem 3.1 is a p-valence version of the univalence criterion in the unit disk obtained earlier by Epstein [8].
If we take in Theorem 3.1, we obtain a p-valence version of Nehari’s [7] univalence criterion.
Corollary 3.2 Let . If
for all , then f is the pth power of a univalent function in .
If we take in Theorem 3.1, we obtain a p-valence version of Beckers’s [5] univalence criterion which was proved in [2].
Corollary 3.3 Let . If
for all , then f is the pth power of a univalent function in .
The following theorem contains another sufficient condition for analytic functions to be univalent in the open unit disk .
Theorem 3.4 Let . If
for all , then the function F is a univalent function in .
Proof Let and . Thus we obtain
It is easy to see that f and g satisfy the assumption of Theorem 3.1 if they satisfy the assumption of this theorem. Thus F is a univalent function in because f in view of Theorem 3.1 is the p th power of a univalent function. □
References
Srivastava HM, Owa S (Eds): Current Topics in Analytic Function Theory. World Scientific, Singapore; 1992.
Hallenbeck, DJ, Livingston, AE: Subordination chains and p-valent functions. Preprint (1975)
Pommerenke C: Über die Subordination analytischer Funktionen. J. Reine Angew. Math. 1965, 218: 159–173.
Pommerenke C: Univalent Functions. Vandenhoeck Ruprecht, Göttingen; 1975.
Becker J: Löwnersche differentialgleichung und quasikonform fortsetzbare schlichte functionen. J. Reine Angew. Math. 1972, 255: 23–43. (in German)
Ahlfors LV: Sufficient conditions for quasiconformal extension. Ann. Math. Stud. 1974, 79: 23–29.
Nehari Z: The Schwarzian derivative and schlicht functions. Bull. Am. Math. Soc. 1949, 55: 545–551. 10.1090/S0002-9904-1949-09241-8
Epstein CL: Univalence criteria and surfaces in hyperbolic space. J. Reine Angew. Math. 1987, 380: 196–214.
Pommerenke C: On the Epstein univalence criterion. Results Math. 1986, 10: 143–146. 10.1007/BF03322371
Deniz, E, Orhan, H, Çağlar, M: Sufficient conditions for p-valence of an integral operator (submitted)
Deniz E, Orhan H: Some notes on extensions of basic univalence criteria. J. Korean Math. Soc. 2011, 48(1):179–189.
Deniz E, Orhan H: Univalence criterion for meromorphic functions and Loewner chains. Appl. Math. Comput. 2011, 218(6):751–755.
Deniz E: Sufficient conditions for univalence and quasiconformal extensions of meromorphic functions. Georgian Math. J. 2012, 19(4):639–653.
Kanas S, Lecko A: Univalence criteria connected with arithmetic and geometric means, II. Folia Sci. Univ. Tech. Resov. 1996, 20: 49–59.
Kanas S, Lecko A: Univalence criteria connected with arithmetic and geometric means, II. In Proceedings of the Second Int. Workshop of Transform Methods and Special Functions. Varna, ’96 Bulgar. Acad. Sci., Sofia; 1996:201–209.
Kanas S, Srivastava HM: Some criteria for univalence related to Ruscheweyh and Salagean derivatives. Complex Var. Elliptic Equ. 1997, 38: 263–275.
Lewandowski Z: On a univalence criterion. Bull. Acad. Pol. Sci., Sér. Sci. Math. 1981, 29: 123–126.
Ovesea H: A generalization of Ruscheweyh’s univalence criterion. J. Math. Anal. Appl. 2001, 258: 102–109. 10.1006/jmaa.2000.7362
Raducanu D, Orhan H, Deniz E: On some sufficient conditions for univalence. An. Univ. ‘Ovidius’ Constanţa, Ser. Mat. 2010, 18(2):217–222.
Ruscheweyh S: An extension of Becker’s univalence condition. Math. Ann. 1976, 220: 285–290. 10.1007/BF01431098
Acknowledgements
Dedicated to Professor Hari M Srivastava.
This project was supported by the Commission for the Scientific Research Projects of Kafkas University. Project number: 2012-FEF-30.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Deniz, E. p-subordination chains and p-valence criteria. J Inequal Appl 2013, 127 (2013). https://doi.org/10.1186/1029-242X-2013-127
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-127