Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volumeย 2011ย (2011), Article IDย 259796, 11 pages
http://dx.doi.org/10.1155/2011/259796
Research Article

Univalent Functions in the Mรถbius Invariant Space

1Departamento de Anรกlisis Matemรกtico, Universidad de La Laguna, Tenerife 38271 La Laguna, Spain
2Department of Physics and Mathematics, University of Eastern Finland, Campus of Joensuu, P.O. Box 111, 80101 Joensuu, Finland

Received 7 July 2011; Accepted 31 August 2011

Academic Editor: Svatoslavย Stanฤ›k

Copyright ยฉ 2011 Fernando Pรฉrez-Gonzรกlez and Jouni Rรคttyรค. 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

It is shown that a univalent function belongs to if and only if , where , provided satisfies certain regularity conditions. It is also shown that under these conditions contains all univalent Bloch functions if and only if .

1. Introduction

The aim of this paper is to characterize univalent functions in the Mรถbius invariant space in terms of the maximum modulus of , where , provided satisfies certain regularity conditions. We will begin with a brief overview of characterizations of univalent functions in classical function spaces of the unit disc together with necessary definitions. Then, we will state the above-mentioned characterization of univalent functions in and its consequences. The proofs are postponed to the end of the paper and will be presented in the sequential order.

2. Notation, Background, and Results

Let denote the algebra of all analytic functions in the unit disc . A function is said to be univalent if it is one to one, and the class of all univalent functions is denoted by . For the theory of univalent functions, see [1โ€“3].

For , the Hardy space consists of those for which where are the standard -means of the restriction of to the circle of radius centered at the origin, and is the maximum modulus function. For the theory of Hardy spaces, see [4, 5].

Hardy and Littlewood [6], Pommerenke [7], and Prawitz [8] proved that if , then , if and only if For more information on univalent functions in Hardy spaces, see [2, 7โ€“13]. A result due to Baernstein II [14] states that , the space of analytic functions in the Hardy space with boundary values of bounded mean oscillation, if and only if , where is the automorphism of which interchanges the origin and the point . This, applied to (2.4), shows that belongs to if and only if for some (equivalently for all) . However, Pommerenke [15] has shown that and the Bloch space contain the same univalent functions. The Bloch space consists of those for which If, in addition, , as , then . For the theory of Bloch spaces, see [3, 16]. If is a simply connected proper subdomain of the complex plane and such that , then where stands for the Euclidean distance from to the boundary of , see, for example, [3]. Therefore, univalent functions in the Bloch space can be characterized by the following well-known geometric condition; if and only if , that is, if the image of under does not contain arbitrarily large discs. For other characterizations of univalent Bloch functions, see [11].

Aulaskari et al. [17] improved the result by Pommerenke by showing that for any . Recall that is a Mรถbius invariant subspace of and consists of those for which where is the Euclidean area element on . In particular, and for all . Moreover, is the classical Dirichlet space which consists of all with finite area of image counting multiplicities. The Dirichlet space is a special case of classical Besov spaces. A geometric characterization of univalent functions in Besov spaces was found by Walsh [18], see also the related results by Donaire et al. [19]. For the theory of spaces, see [20, 21].

For , the Mรถbius invariant space consists of those for which Moreover, if the integral above tends to zero as approaches to the boundary of , then . If , then , and therefore can be viewed as a generalized space. For results on , see [22โ€“24] and the references therein. From now on, the weight is assumed to admit the following basic properties: (1);(2);(3) for all ;(4) for all .

Requirements (1) and (2) are standard; the first one ensures that indeed plays an essential role in the definition, and the second one guarantees the nontriviality of as well as the inclusions and . Conditions (3) and (4) are, of course, restrictions, yet, for example, satisfies both of them for all .

Before proceeding further, we give an example related to spaces in order to illustrate the variety of spaces induced by different choices of .

Example 2.1. For and , consider the weight Obviously, for all . Since for either (the case ) or (the case ), the weight is increasing. If , then the integral diverges for all , and therefore by [22, Theorem 2.3]. Moreover, [22, Theorem 2.6] shows that for all and .

Let us turn back to univalent functions. A special case of [11, Theorem 4] shows that a univalent function belongs to if and only if Of course, this is not a natural way to state the result because for all , but it appears to be useful for our purposes. The case of (2.14), which corresponds to, reduces to (2.5) with . An appropriate interpretation of (2.14) allows us to conclude what happens with . Namely, since for , condition (2.14), with being replaced by , gives a candidate for a characterization of univalent functions in . The main result of this paper is Theorem 2.2 which shows that this is indeed the case, provided for some positive constant .

Theorem 2.2. Let and assume that satisfies condition (2.15). Then, if and only if Moreover, if and only if

It is an immediate consequence of the proof that the assertions in Theorem 2.2 remain valid for areally mean -valent functions. In addition to standard techniques of univalent functions, the proof of Theorem 2.2 uses a result by Pavloviฤ‡ and Pelรกez [25] on weighted integrals of analytic functions and their derivatives. An application of this result yields the requirement (2.15). We will analyze the importance of (2.15) after discussing consequences of Theorem 2.2 and its proof.

The first part of the proof of Theorem 2.2 shows that belongs to if and only if where denotes the area of image of the pseudohyperbolic disc under counting multiplicities. This, together with Theorem 2.2, shows that for , the quantities and are of the same growth (uniformly in ) when measured in terms of .

Since for any by a result due to Aulaskari et al. [17], it is natural to ask when does the identity hold. An application of Theorem 2.2 yields Corollary 2.3 which answers this question provided satisfies (2.15).

Corollary 2.3. Assume that satisfies condition (2.15). Then, if and only if Moreover, if (2.19) is satisfied, then .

Wulan [26] showed that every areally mean -valent Bloch function belongs to whenever satisfies

This result is more general than Corollary 2.3 because the crucial condition (2.15) is not needed. However, is concave whenever (4) is satisfied, and in that case, , and then (2.20) implies (2.19).

We next analyze the necessity of condition (2.15). This is needed in two consecutive steps in the proof of Theorem 2.2. These steps together establish the asymptotic inequality for all . As usual, we write , if there exists such that , and the notation is understood in an analogous manner. In particular, if and , then we write . Set now , so that (2.21) becomes We will show next that (2.22) does not necessarily remain true even for univalent functions , if , that induces , does not satisfy (2.15). To see this, take and with , so that For these choices, the right-hand side of (2.22) is finite, whereas the left-hand side is not. If , then , but (2.15) for this is equivalent to , which clearly fails as . Moreover, , but (2.19) fails for . This justifies the assumption (2.15) in Theorem 2.2 and Corollary 2.3.

3. Proof of Theorem 2.2

We will prove the first assertion, the second one, then immediately follows by the proof.

3.1. Sufficiency of (2.16)

Let . Recall first that is nondecreasing for any . Therefore we may use conditions (1) and (3) together with Fubiniโ€™s theorem to obtain If , then and an application of this inequality to , together, with (3.1), gives Therefore a univalent function belongs to if (2.16) is satisfied. Note that this part of the proof uses the univalence of , but does not rely on (2.15). Moreover, if is areally mean -valent, then the reasoning in (3.2) remains valid as soon as the right-hand side of the inequality is multiplied by .

3.2. Necessity of (2.16)

To prove this, we will need a special case of a result due to Pavloviฤ‡ and Pelรกez [25], which states that provided is differentiable and satisfies for some constant . Assumption (2.15) is equivalent to (3.5) for . Therefore, (3.4) yields We next show that where is the constant in (2.15). To prove (3.7), consider the function By the assumptions (1), (3), and (4) on , as . Moreover, (2.15) yields and thus is decreasing, and therefore (3.7) holds. This, together with Fubini's theorem, shows that from which the inequality , valid for all (see [9, page 841] or [7, Hilfssatz 1]), yields Thus, (2.16) is satisfied if .

4. Proof of Corollary 2.3

4.1. Sufficiency of (2.19)

Since , it suffices to show that . To see this, let . An application of the inequality to the function gives for all . This yields and thus by Theorem 2.2.

4.2. Necessity of (2.19)

If satisfies condition (2.15) and , then the univalent Bloch function , together with Theorem 2.2, shows that (2.19) holds.

4.3. The Case

Assume that satisfies (2.19) and let . Since , it suffices to show that . To see this, let . By applying (4.2) we obtain Since , the assumption (2.19) implies that for all sufficiently large. Fix such an . The image of the circle of radius centered at the origin under a univalent function is a Jordan curve with zero in its inner domain. The length of this image is , and hence . This estimate, applied to , together with the assumption , yields for all sufficiently large . Therefore, and hence by Theorem 2.2.

Acknowledgments

This research was supported in part by the MEC-Spain MTM2008-05891, the Academy of Finland 121281, and the European Networking Programme HCAA of the European Science Foundation.

References

  1. P. L. Duren, Univalent Functions, vol. 259, Springer, New York, NY, USA, 1983.
  2. C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, Germany, 1975.
  3. C. Pommerenke, Boundary Behaviour of Conformal Maps, vol. 299, Springer, Berlin, Germany, 1992.
  4. P. L. Duren, Theory of Hp Spaces, Academic Press, New York, NY, USA, 1970.
  5. J. B. Garnett, Bounded Analytic Functions, vol. 96 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1981.
  6. G. H. Hardy and J. E. Littlewood, โ€œSome properties of fractional integrals. II,โ€ Mathematische Zeitschrift, vol. 34, no. 1, pp. 403โ€“439, 1932. View at Publisher ยท View at Google Scholar
  7. C. Pommerenke, โ€œÜber die Mittelwerte und Koeffizienten multivalenter Funktionen,โ€ Mathematische Annalen, vol. 145, pp. 285โ€“296, 1961/1962. View at Google Scholar
  8. H. Prawitz, โ€œUber Mittelwerte analytischer Funktionen,โ€ Arkiv för Matematik, Astronomi och Fysik, vol. 20, pp. 1โ€“12, 1927. View at Google Scholar
  9. A. Baernstein II, D. Girela, and J. A. Peláez, โ€œUnivalent functions, Hardy spaces and spaces of Dirichlet type,โ€ Illinois Journal of Mathematics, vol. 48, no. 3, pp. 837โ€“859, 2004. View at Google Scholar
  10. D. Girela, M. Pavlović, and J. A. Peláez, โ€œSpaces of analytic functions of Hardy-Bloch type,โ€ Journal d'Analyse Mathématique, vol. 100, pp. 53โ€“81, 2006. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  11. F. Pérez-González and J. Rättyä, โ€œUnivalent functions in Hardy, Bergman, Bloch and related spaces,โ€ Journal d'Analyse Mathématique, vol. 105, pp. 125โ€“148, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  12. P. Galanopoulos, D. Girela, and R. Hernández, โ€œUnivalent functions, VMOA and related spaces,โ€ Journal of Geometric Analysis, vol. 21, no. 3, pp. 665โ€“682, 2011. View at Google Scholar
  13. C. González and J. A. Peláez, โ€œUnivalent functions in Hardy spaces in terms of the growth of arc-length,โ€ Journal of Geometric Analysis, vol. 19, no. 4, pp. 755โ€“771, 2009. View at Google Scholar
  14. A. Baernstein II, โ€œAnalytic functions of bounded mean oscillation,โ€ in Aspects of Contemporary Complex Analysis, D. Brannan and J. Clunie, Eds., pp. 3โ€“36, Academic Press, London, UK, 1980. View at Google Scholar ยท View at Zentralblatt MATH
  15. C. Pommerenke, โ€œSchlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation,โ€ Commentarii Mathematici Helvetici, vol. 52, no. 4, pp. 591โ€“602, 1977. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  16. K. H. Zhu, Operator Theory in Function Spaces, vol. 139 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1990.
  17. R. Aulaskari, P. Lappan, J. Xiao, and R. Zhao, โ€œOn α-Bloch spaces and multipliers of Dirichlet spaces,โ€ Journal of Mathematical Analysis and Applications, vol. 209, no. 1, pp. 103โ€“121, 1997. View at Publisher ยท View at Google Scholar
  18. D. Walsh, โ€œA property of univalent functions in Ap,โ€ Glasgow Mathematical Journal, vol. 42, no. 1, pp. 121โ€“124, 2000. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  19. J. J. Donaire, D. Girela, and D. Vukotić, โ€œOn univalent functions in some Möbius invariant spaces,โ€ Journal für die Reine und Angewandte Mathematik, vol. 553, pp. 43โ€“72, 2002. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  20. J. Xiao, Holomorphic Q Classes, vol. 1767 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2001. View at Publisher ยท View at Google Scholar
  21. J. Xiao, Geometric Qp Functions, Frontiers in Mathematics, Birkhäuser, Basel, Switzerland, 2006. View at Zentralblatt MATH
  22. M. Essén and H. Wulan, โ€œOn analytic and meromorphic functions and spaces of QK-type,โ€ Illinois Journal of Mathematics, vol. 46, no. 4, pp. 1233โ€“1258, 2002. View at Google Scholar
  23. M. Essén, H. Wulan, and J. Xiao, โ€œSeveral function-theoretic characterizations of Möbius invariant QK spaces,โ€ Journal of Functional Analysis, vol. 230, no. 1, pp. 78โ€“115, 2006. View at Google Scholar ยท View at Zentralblatt MATH
  24. H. Wulan and P. Wu, โ€œCharacterizations of QT spaces,โ€ Journal of Mathematical Analysis and Applications, vol. 254, no. 2, pp. 484โ€“497, 2001. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  25. M. Pavlović and J. A. Peláez, โ€œAn equivalence for weighted integrals of an analytic function and its derivative,โ€ Mathematische Nachrichten, vol. 281, no. 11, pp. 1612โ€“1623, 2008. View at Publisher ยท View at Google Scholar
  26. H. Wulan, โ€œMultivalent functions and QK spaces,โ€ International Journal of Mathematics and Mathematical Sciences, no. 45–48, pp. 2537โ€“2546, 2004. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH