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Univalence criteria for meromorphic functions and quasiconformal extensions
Journal of Inequalities and Applications volume 2013, Article number: 112 (2013)
The aim of this paper is to obtain sufficient conditions for univalence of meromorphic functions in the . Also, we refine a quasiconformal extension criterion with the help of Becker’s method. A number of univalence conditions would follow upon specializing the parameters involved in our main results.
MSC:30C80, 30C45, 30C62.
We denote by () the disc of radius r and let . Let denote the class of analytic functions in the open unit disc which satisfy the usual normalization condition . We denote by the subclass of consisting of functions which are univalent in . Let Σ denote the class of functions of the form which are meromorphic in the exterior to the open unit disc with a pole at the infinity residue 1. We say that a sense-preserving homeomorphism f of a plane domain is k-quasiconformal if f is absolutely continuous on almost all lines parallel to coordinate axes and , almost everywhere in G, where , and k is a constant with .
In geometric function theory, the univalence of complex functions is an important property, but it is difficult, and in many cases impossible, to show directly that a certain complex function is univalent. For this reason, many authors found different types of sufficient conditions of univalence. One of the most important of these conditions of univalence in the domains and the exterior of a closed unit disc is the well-known criterion of Becker [1, 2]. Becker’s work depends upon a clever use of the theory of Loewner chains and the generalized Loewner differential equation. Extensions of this criterion were given by Ahlfors , Lewandowski [4, 5], Miazga and Wesolowski , Ruscheweyh , and Singh and Chichra . Also, the recent investigations on this subject by Deniz and Orhan [9–11], Ponnusamy and Sugawa , Răducanu et al. , Kanas and Lecko [14, 15] and Kanas and Srivastava .
In the present paper, firstly we study a number of new criteria for the univalence of the functions belonging to the class Σ. Finally, we obtain a refinement to a quasiconformal extension criterion of the main result. We also consider several interesting corollaries and consequences of our univalence criteria. Our considerations are based on the theory of Loewner chains.
2 Loewner chains and quasiconformal extensions
Before proving our main theorem, we need a brief summary of the method of Loewner chains.
Let , be a function defined on , where and is a complex-valued, locally absolutely continuous function on I. is called a Loewner chain if satisfies the following conditions:
is analytic and univalent in for all ,
for all ,
where the symbol ‘≺’ stands for subordination. If , then we say that is a standard Loewner chain.
In order to prove our main results, we need the following theorem due to Pommerenke  (also see ). This theorem is often used to find out univalency for an analytic function, apart from the theory of Loewner chains.
Theorem 2.1 (see Pommerenke )
Let be analytic in for all . Suppose that
is a locally absolutely continuous function in the interval I, and locally uniform with respect to .
is a complex-valued continuous function on I such that , for and
forms a normal family of functions in .
There exists an analytic function satisfying for all , and(2.1)
Then, for each , the function has an analytic and univalent extension to the whole disc or the function is a Loewner chain.
Equation (2.1) is called the generalized Loewner differential equation.
Theorem 2.2 Suppose that is a Loewner chain for which in (2.1) satisfies the condition
for all and . Then admits a continuous extension to for each , and the function defined by
is a k-quasiconformal extension of to ℂ.
3 Univalence criteria
Making use of Theorem 2.1, now we can prove our main result.
Theorem 3.1 Let and s be a complex number such that , , ; and , . If there exists an analytic function g in such that and the inequalities
hold true for all , then the function f is univalent in .
Proof We will prove that there exists a real number such that the function , defined formally by
is analytic in for all , where
Since g is analytic, the function
is analytic in and . Then there exists a disc , , in which for all . We denote the uniform branch of by , which is equal to at origin.
It follows from (3.3) that
and thus the function is analytic in .
for which we consider the uniform branch equal to at the origin.
Because and , we have
Moreover, for all .
After simple calculation, we obtain, for each ,
The limit function belongs to the family ; then in every closed disc , , there exists a constant such that
uniformly in this disc, provided that t is sufficiently large. Then, by Montel’s theorem, is a normal family in . From the analyticity of , we obtain that for all fixed numbers and , , there exists a constant (that depends on T and ) such that
Therefore, the function is locally absolutely continuous in I, locally uniform with respect to .
The function defined by
is analytic in a disc , , for all .
If the function
is analytic in and for all and , then has an analytic extension with a positive real part in for all . We take ζ instead of z in equality (3.6); then we have
for and .
The inequality for all and is equivalent to
where is defined by (3.7).
From (3.1), (3.8) and (3.10), we have
Since for all and , we find that is an analytic function in . Using the maximum modulus principle, it follows that for all and each , arbitrarily fixed, there exists such that
for all and .
Denote . Then , and from (3.8) we have
Because , inequality (3.2) implies that
and from (3.11), (3.13) and (3.12), we conclude that
for all and . Therefore for all and .
Since all the conditions of Theorem 2.1 are satisfied, we obtain that the function has an analytic and univalent extension to the whole unit disc , for all . For , we have , for and therefore the function is univalent in . □
If we take in Theorem 3.1, then we have the following result.
Corollary 3.2 Let and s be a complex number such that , , ; and , . If the inequality
holds true for all , then the function f is univalent in .
For in Theorem 3.1, we have the following corollary.
Corollary 3.3 Let and s be a complex number such that , , ; and , . If the inequalities
hold true for all , then the function f is univalent in .
Putting in Theorem 3.1, we obtain a simple univalence condition as follows.
Corollary 3.4 Let and s be a complex number such that , , ; and , . If the inequality
holds true for all , then the function f is univalent in .
4 Quasiconformal extension criterion
In this section we will refine the univalence condition given in Theorem 3.1 to a quasiconformal extension criterion.
Theorem 4.1 Let and s be a complex number such that , , ; ; and , . If there exists an analytic function g in such that and the inequalities
hold true for all , then the function f has a l-quasiconformal extension to ℂ, where
Proof In the proof of Theorem 3.1, it has been shown that the function given by (3.3) is a subordination chain in . Applying Theorem 2.2 to the function given by (3.7), we obtain that the condition
implies l-quasiconformal extensibility of f, where is defined by (3.8).
Lengthy but elementary calculation shows that the last inequality (4.3) is equivalent to
It is easy to check that, under the assumptions (4.1) and (4.2), we have
Consider the two discs Δ and defined by (4.4) and (4.5) respectively, where is replaced by a complex variable w. Our theorem will be proved if we find the smallest for which is contained in Δ. This will be done if and only if the distance apart of the centers plus the smallest radius is equal, at most, to the largest radius. So, we are required to prove that
with the condition
Now we will solve inequalities (4.6) and (4.7). If in (4.6) the inequality sign is replaced by equal, making use of Mathematica program, we obtain the following two solutions:
Therefore, the solution of inequality (4.6) is and . Since , it remains .
After similar calculations, from inequality (4.7), we have and , where
Since , we get .
Again, making use of Mathematica program, we obtain . Therefore and the proof is complete. □
Becker J: Löwnersche differential gleichung und quasikonform fortsetzbare schlichte functionen. J. Reine Angew. Math. 1972, 255: 23–43. (in German)
Becker J: Über die Lösungsstruktur einer Differentialgleichung in der Konformen Abbildung. J. Reine Angew. Math. 1976, 285: 66–74.
Ahlfors LV: Sufficient conditions for quasiconformal extension. Ann. Math. Stud. 1974, 79: 23–29.
Lewandowski Z: On a univalence criterion. Bull. Acad. Pol. Sci., Sér. Sci. Math. 1981, 29: 123–126.
Lewandowski Z: Some remarks on univalence criteria. Ann. Univ. Mariae Curie-Skl̄odowska, Sect. A 1982/1983, 36/37: 87–95.
Miazga J, Wesolowski A: A univalence criterion for meromorphic functions. Ann. Pol. Math. 1991, 56(1):63–66.
Ruscheweyh S: An extension of Becker’s univalence condition. Math. Ann. 1976, 220: 285–290. 10.1007/BF01431098
Singh V, Chichra PN: An extension of Becker’s criterion for univalence. J. Indian Math. Soc. 1977, 41: 353–361.
Deniz E: Sufficient conditions for univalence and quasiconformal extensions of meromorphic functions. Georgian Math. J. 2012, 19(4):639–653.
Deniz E, Orhan H: Univalence criterion for analytic functions. Gen. Math. 2009, 4: 211–220.
Deniz E, Orhan H: Univalence criterion for meromorphic functions and Loewner chains. Appl. Math. Comput. 2011, 218(6):751–755.
Ponnusamy S, Sugawa T: Norm estimates and univalence criteria for meromorphic functions. J. Korean Math. Soc. 2008, 45(6):1661–1676. 10.4134/JKMS.2008.45.6.1661
Răducanu D, Orhan H, Deniz E: On some sufficient conditions for univalence. An. Univ. ‘Ovidius’ Constanţa, Ser. Mat. 2010, 18(2):217–222.
Kanas S, Lecko A: Univalence criteria connected with arithmetic and geometric means, II. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 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.
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: Conformal mappings with quasiconformal extensions. In Aspects of Contemporary Complex Analysis. Edited by: Brannan DA, Clunie JG. Academic Press, San Diego; 1980:37–77.
Anderson JM, Hinkkanen A: Univalence criteria and quasiconformal extensions. Trans. Am. Math. Soc. 1991, 324: 823–842. 10.1090/S0002-9947-1991-0994162-4
Betker Th: Löewner chains and quasiconformal extensions. Complex Var. 1992, 20: 107–111. 10.1080/17476939208814591
Pfaltzgraff JA: k -quasiconformal extension criteria in the disk. Complex Var. 1993, 21: 293–301. 10.1080/17476939308814638
Hotta I: Löwner chains with complex leading coefficient. Monatshefte Math. 2011, 163(3):315–325. 10.1007/s00605-010-0200-5
Hotta I: Explicit quasiconformal extensions and Loewner chains. Proc. Jpn. Acad., Ser. A, Math. Sci. 2009, 85: 108–111.
Hotta, I: Löewner chains and quasiconformal extension of univalent functions, Dissertation, Tohoku Univ., 2010
Dedicated to Professor Hari M Srivastava.
The authors thank the referees for their valuable suggestions to improve the paper. The present investigation was supported by Atatürk University Rectorship under ‘The Scientific and Research Project of Atatürk University’, Project No: 2012/173.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Çağlar, M., Orhan, H. Univalence criteria for meromorphic functions and quasiconformal extensions. J Inequal Appl 2013, 112 (2013). https://doi.org/10.1186/1029-242X-2013-112
- univalent function
- meromorphic function
- quasiconformal extension
- univalence condition
- Loewner chain