Advertisement

A Survey on Blaschke-Oscillatory Differential Equations, with Updates

  • Janne HeittokangasEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 65)

Abstract

In the celebrated 1949 paper due to Nehari, necessary and sufficient conditions are given for a locally univalent meromorphic function to be univalent in the unit disc \(\mathbb{D}\). The proof involves a second order differential equation of the form
$$ f''+A(z)f=0, $$
(†)
where A(z) is analytic in \(\mathbb{D}\). As an immediate consequence of the proof, it follows that if |A(z)|≤1/(1−|z|2)2 for every \(z\in\mathbb{D}\), then any non-trivial solution of () has at most one zero in \(\mathbb{D}\).

Since 1949 a number of papers provide with different types of growth conditions on the coefficient A(z) such that the solutions of () have at most finitely many zeros in \(\mathbb{D}\). If there exists at least one solution with infinitely many zeros in \(\mathbb{D}\), then () is oscillatory. If the zeros still satisfy the classical Blaschke condition, then () is called Blaschke-oscillatory. This concept was introduced by the author in 2005, but the topic was considered by Hartman and Wintner already in 1955 (Trans. Am. Math. Soc. 78:492–500). This semi-survey paper provides with a collection of results and tools dealing with Blaschke-oscillatory equations.

As for results, necessary and sufficient conditions are given, and notable effort has been put in dealing with prescribed zero sequences satisfying the Blaschke condition. The concept of Blaschke-oscillation also extends to differential equations of arbitrary order. Many of the results given in this paper have been published earlier in a weaker form. All questions regarding the zeros of solutions can be rephrased for the critical points of solutions. This gives rise to a new concept called Blaschke-critical equations. To intrigue the reader, several open problems are pointed out in the text.

Some classical tools and closely related topics that are often related to the finite oscillation case include the Schwarzian derivative, properties of univalent functions, Green’s identity, conformal mappings, and a certain Hardy-Littlewood inequality. The Blaschke-oscillatory case also makes use of interpolation theory, various growth estimates for logarithmic derivatives of Blaschke products, Bank-Laine functions and recently updated Wiman-Valiron theory.

Keywords

Blaschke-critical Blaschke-oscillatory Blaschke product Differential equation Logarithmic derivative Oscillation theory Prescribed zeros Zero sequence 

Mathematics Subject Classification

34M10 30J10 30H15 

References

  1. 1.
    Aulaskari, R., Nowak, M., Zhao, R.: The nth derivative characterisation of Möbius invariant Dirichlet space. Bull. Aust. Math. Soc. 58(1), 43–56 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aulaskari, R., Xiao, J., Zhao, R.: On subspaces and subsets of BMOA and UBC. Analysis 15(2), 101–121 (1995) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bank, S.: A general theorem concerning the growth of solutions of first-order algebraic differential equations. Compos. Math. 25, 61–70 (1972) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bank, S., Laine, I.: On the oscillation theory of f″+Af=0 where A is entire. Trans. Am. Math. Soc. 273(1), 351–363 (1982) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Buckley, S., Koskela, P., Vukotic, D.: Fractional integration, differentiation, and weighted Bergman spaces. Math. Proc. Camb. Philos. Soc. 126(2), 369–385 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Carmona, J., Cufi, J., Pommerenke, Ch.: On the angular limits of Bloch functions. Publ. Mat. 32(2), 191–198 (1988) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chuaqui, M., Duren, P., Osgood, B.: Schwarzian derivatives of convex mappings. Ann. Acad. Sci. Fenn. Math. 36, 449–460 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chuaqui, M., Duren, P., Osgood, B., Stowe, D.: Oscillation of solutions of linear differential equations. Bull. Aust. Math. Soc. 79(1), 161–169 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chuaqui, M., Stowe, D.: Valence and oscillation of functions in the unit disk. Ann. Acad. Sci. Fenn. Math. 33(2), 561–584 (2008) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chyzhykov, I., Gundersen, G.G., Heittokangas, J.: Linear differential equations and logarithmic derivative estimates. Proc. Lond. Math. Soc. 86(3), 735–754 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chyzhykov, I., Heittokangas, J., Rättyä, J.: Finiteness of φ-order of solutions of linear differential equations in the unit disc. J. Anal. Math. 109(1), 163–198 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Cima, J., Colwell, P.: Blaschke quotients and normality. Proc. Am. Math. Soc. 19, 796–798 (1968) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Cima, J., Pfaltzgraff, J.: Oscillatory behavior of u″(z)+h(z)u(z)=0 for univalent h(z). J. Anal. Math. 25, 311–322 (1972) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Clunie, J.: The derivative of a meromorphic function. Proc. Am. Math. Soc. 7, 227–229 (1956) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Duren, P.: Theory of H p Spaces. Academic Press, New York (1970) zbMATHGoogle Scholar
  16. 16.
    Duren, P., Romberg, B., Shields, A.: Linear functionals on H p spaces with 0<p<1. J. Reine Angew. Math. 238, 32–60 (1969) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Duren, P., Schuster, A.: Bergman Spaces. Mathematical Surveys and Monographs, vol. 100. Am. Math. Soc., Providence (2004) zbMATHGoogle Scholar
  18. 18.
    Essén, M., Xiao, J.: Some results on Q p spaces, 0<p<1. J. Reine Angew. Math. 485, 173–195 (1997) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fenton, P., Rossi, J.: ODEs and Wiman-Valiron theory in the unit disc. J. Math. Anal. Appl. 367(1), 137–145 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Fricain, E., Mashreghi, J.: Exceptional sets for the derivatives of Blaschke products. In: Proceedings of the St. Petersburg Mathematical Society. Vol. XIII. Amer. Math. Soc. Transl. Ser. 2, vol. 222, pp. 163–170. Am. Math. Soc., Providence (2008) Google Scholar
  21. 21.
    Fricain, E., Mashreghi, J.: Integral means of the derivatives of Blaschke products. Glasg. Math. J. 50(2), 233–249 (2008) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Girela, D., Peláez, J., Pérez-González, F., Rättyä, J.: Carleson measures for the Bloch space. Integral Equ. Oper. Theory 61(4), 511–547 (2008) zbMATHCrossRefGoogle Scholar
  23. 23.
    Girela, D., Peláez, J., Vukotic’, D.: Integrability of the derivative of a Blaschke product. Proc. Edinb. Math. Soc. (2) 50(3), 673–687 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Girela, D., Peláez, J., Vukotic’, D.: Uniformly discrete sequences in regions with tangential approach to the unit circle. Complex Var. Elliptic Equ. 52(2–3), 161–173 (2007) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Gnuschke-Hauschild, D., Pommerenke, Ch.: On Bloch functions and gap series. J. Reine Angew. Math. 367, 172–186 (1986) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Gotoh, Y.: On integral means of the derivatives of Blaschke products. Kodai Math. J. 30(1), 147–155 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Gröhn, J., Heittokangas, J.: New findings on Bank-Sauer approach in oscillation theory. Constr. Approx. 35, 345–361 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Hartman, P., Wintner, A.: On linear second order differential equations in the unit circle. Trans. Am. Math. Soc. 78, 492–500 (1955) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Hayman, W.K.: Meromorphic Functions. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964) zbMATHGoogle Scholar
  30. 30.
    Hayman, W.K.: On the characteristic of functions meromorphic in the unit disk and of their integrals. Acta Math. 112, 181–214 (1964) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Hayman, W.K.: Multivalent Functions, 2nd edn. Cambridge Tracts in Mathematics, vol. 110. Cambridge University Press, Cambridge (1994) zbMATHCrossRefGoogle Scholar
  32. 32.
    Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Graduate Texts in Mathematics, vol. 199. Springer, New York (2000) zbMATHCrossRefGoogle Scholar
  33. 33.
    Heittokangas, J.: On complex differential equations in the unit disc. Ann. Acad. Sci. Fenn. Math. Diss. 122, 1–54 (2000) MathSciNetGoogle Scholar
  34. 34.
    Heittokangas, J.: Solutions of f″+A(z)f=0 in the unit disc having Blaschke sequences as the zeros. Comput. Methods Funct. Theory 5(1), 49–63 (2005) MathSciNetzbMATHGoogle Scholar
  35. 35.
    Heittokangas, J.: Blaschke-oscillatory equations of the form f″+A(z)f=0. J. Math. Anal. Appl. 318(1), 120–133 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Heittokangas, J.: Growth estimates for logarithmic derivatives of Blaschke products and of functions in the Nevanlinna class. Kodai Math. J. 30, 263–279 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Heittokangas, J.: On interpolating Blaschke products and Blaschke-oscillatory equations. Constr. Approx. 34(1), 1–21 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Heittokangas, J., Korhonen, R., Rättyä, J.: Growth estimates for solutions of linear complex differential equations. Ann. Acad. Sci. Fenn. 29(1), 233–246 (2004) MathSciNetzbMATHGoogle Scholar
  39. 39.
    Heittokangas, J., Korhonen, R., Rättyä, J.: Linear differential equations with solutions in the Dirichlet type subspace of the Hardy space. Nagoya Math. J. 187, 91–113 (2007) MathSciNetzbMATHGoogle Scholar
  40. 40.
    Heittokangas, J., Korhonen, R., Rättyä, J.: Linear differential equations with coefficients in weighted Bergman and Hardy spaces. Trans. Am. Math. Soc. 360(2), 1035–1055 (2008) zbMATHCrossRefGoogle Scholar
  41. 41.
    Heittokangas, J., Korhonen, R., Rättyä, J.: Growth estimates for solutions of nonhomogeneous linear differential equations. Ann. Acad. Sci. Fenn. 34(1), 145–156 (2009) MathSciNetzbMATHGoogle Scholar
  42. 42.
    Heittokangas, J., Laine, I.: Solutions of f″+A(z)f=0 with prescribed sequences of zeros. Acta Math. Univ. Comen. 124(2), 287–307 (2005) MathSciNetGoogle Scholar
  43. 43.
    Heittokangas, J., Rättyä, J.: Zero distribution of solutions of complex linear differential equations determines growth of coefficients. Math. Nachr. 284(4), 412–420 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Heittokangas, J., Tohge, K.: A unit disc analogue of the Bank-Laine conjecture does not hold. Ann. Acad. Sci. Fenn. 36(1), 341–351 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Herold, H.: Nichteuklidischer Nullstellenabstand der Lösungen von w″+p(z)w=0. Math. Ann. 287, 637–642 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Hille, E.: Remarks on a paper by Zeev Nehari. Bull. Am. Math. Soc. 55, 552–553 (1949) MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Horowitz, C.: Factorization theorems for functions in the Bergman spaces. Duke Math. J. 44(1), 201–213 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Ince, E.: Ordinary Differential Equations. Dover, New York (1956) Google Scholar
  49. 49.
    Juneja, O., Kapoor, G.: Analytic Functions—Growth Aspects. Research Notes in Mathematics, vol. 104. Pitman Adv. Publ. Prog., Boston (1985) zbMATHGoogle Scholar
  50. 50.
    Kim, H.: Derivatives of Blaschke products. Pac. J. Math. 114(1), 175–190 (1984) zbMATHGoogle Scholar
  51. 51.
    Kim, W.: The Schwarzian derivative and multivalence. Pac. J. Math. 31, 717–724 (1969) zbMATHGoogle Scholar
  52. 52.
    Korenblum, B.: An extension of the Nevanlinna theory. Acta Math. 135(3–4), 187–219 (1975) MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Korhonen, R., Rättyä, J.: Finite order solutions of linear differential equations in the unit disc. J. Math. Anal. Appl. 349(1), 43–54 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Laine, I.: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993) CrossRefGoogle Scholar
  55. 55.
    Laine, I.: Complex differential equations. In: Handbook of Differential Equations: Ordinary Differential Equations. Vol. IV. Handb. Differ. Equ., pp. 269–363. Elsevier/North-Holland, Amsterdam (2008) (English summary) Google Scholar
  56. 56.
    Linden, C.: H p-derivatives of Blaschke products. Mich. Math. J. 23, 43–51 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    London, D.: On the zeros of solutions of w″(z)+p(z)w(z)=0. Pac. J. Math. 12, 979–991 (1962) MathSciNetzbMATHGoogle Scholar
  58. 58.
    Mashreghi, J., Shabankhah, M.: Integral means of the logarithmic derivative of Blaschke products. Comput. Methods Funct. Theory 9(2), 421–433 (2009) MathSciNetzbMATHGoogle Scholar
  59. 59.
    Nehari, Z.: The Schwarzian derivative and Schlicht functions. Bull. Am. Math. Soc. 55, 545–551 (1949) MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Nehari, Z.: Conformal Mapping. Dover, New York (1975). Reprinting of the 1952 edn. Google Scholar
  61. 61.
    Nevanlinna, R.: Analytic Functions. Die Grundlehren der mathematischen Wissenschaften, vol. 162. Springer, New York (1970). Translated from the second German edn. by Phillip Emig zbMATHGoogle Scholar
  62. 62.
    Nolder, C.: An L p definition of interpolating Blaschke products. Proc. Am. Math. Soc. 128(6), 1799–1806 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Pascuas, D.: A note on interpolation by Bloch functions. Proc. Am. Math. Soc. 135(7), 2127–2130 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Peláez, J.: Sharp results on the integrability of the derivative of an interpolating Blaschke product. Forum Math. 20(6), 1039–1054 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Peláez, J., Rättyä, J.: Weighted Bergman spaces induced by rapidly increasing weights. Mem. Am. Math. Soc. (to appear) Google Scholar
  66. 66.
    Pommerenke, Ch.: Univalent Functions. Studia Mathematica/Mathematische Lehrbücher, vol. XXV. Vandenhoeck & Ruprecht, Göttingen (1975). With a chapter on quadratic differentials by Gerd Jensen zbMATHGoogle Scholar
  67. 67.
    Pommerenke, Ch.: On the mean growth of the solutions of complex linear differential equations in the disk. Complex Var. Theory Appl. 1(1), 23–38 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Protas, D.: Blaschke products with derivative in H p and B p. Mich. Math. J. 20, 393–396 (1973) MathSciNetzbMATHGoogle Scholar
  69. 69.
    Rättyä, J.: Linear differential equations with solutions in Hardy spaces. Complex Var. Elliptic Equ. 52(9), 785–795 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Schwarz, B.: Complex nonoscillation theorems and criteria of univalence. Trans. Am. Math. Soc. 80, 159–186 (1955) zbMATHCrossRefGoogle Scholar
  71. 71.
    Šeda, V.: A note to a paper by Clunie. Acta Fac. Nat. Univ. Comen. 4, 255–260 (1959) zbMATHGoogle Scholar
  72. 72.
    Šeda, V.: On some properties of solutions of the differential equation y″=Q(z)y, where Q(z)≠0 is an entire function. Acta Fac. Nat. Univ. Comen. Math. 4, 223–253 (1959) (Slovak) zbMATHGoogle Scholar
  73. 73.
    Shea, D., Sons, L.: Value distribution theory for meromorphic functions of slow growth in the disk. Houst. J. Math. 12(2), 249–266 (1986) MathSciNetzbMATHGoogle Scholar
  74. 74.
    Tse, K.-F.: Nontangential interpolating sequences and interpolation by normal functions. Proc. Am. Math. Soc. 29, 351–354 (1971) MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Tsuji, M.: Potential Theory in Modern Function Theory. Chelsea, New York (1975). Reprinting of the 1959 edn. zbMATHGoogle Scholar
  76. 76.
    Yamashita, S.: Gap series and α-Bloch functions. Yokohama Math. J. 28, 31–36 (1980) MathSciNetzbMATHGoogle Scholar
  77. 77.
    Zabulionis, A.: Separation of points of the unit disc. Lith. Math. J. 23(3), 271–274 (1983) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Physics and MathematicsUniversity of Eastern FinlandJoensuuFinland

Personalised recommendations