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On non-normal solutions of linear differential equations


Author: Janne Gröhn
Journal: Proc. Amer. Math. Soc. 145 (2017), 1209-1220
MSC (2010): Primary 34C10
DOI: https://doi.org/10.1090/proc/13292
Published electronically: September 8, 2016
MathSciNet review: 3589320
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Abstract: Normality arguments are applied to study the oscillation of solutions of $ f''+Af=0$, where the coefficient $ A$ is analytic in the unit disc $ \mathbb{D}$ and $ \sup _{z\in \mathbb{D}} (1-\vert z\vert^2)^2\vert A(z)\vert < \infty $. It is shown that such a differential equation may admit a non-normal solution having prescribed uniformly separated zeros.


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  • [1] Rauno Aulaskari and Jouni Rättyä, Properties of meromorphic $ \phi $-normal functions, Michigan Math. J. 60 (2011), no. 1, 93-111. MR 2785866, https://doi.org/10.1307/mmj/1301586306
  • [2] Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547-559. MR 0141789
  • [3] Martin Chuaqui, Janne Gröhn, Janne Heittokangas, and Jouni Rättyä, Zero separation results for solutions of second order linear differential equations, Adv. Math. 245 (2013), 382-422. MR 3084433, https://doi.org/10.1016/j.aim.2013.06.022
  • [4] Joseph A. Cima and Glenn Schober, Analytic functions with bounded mean oscillation and logarithms of $ H^{p}$ functions, Math. Z. 151 (1976), no. 3, 295-300. MR 0425128
  • [5] Flavia Colonna, Bloch and normal functions and their relation, Rend. Circ. Mat. Palermo (2) 38 (1989), no. 2, 161-180. MR 1029707, https://doi.org/10.1007/BF02843992
  • [6] S. M. Elzaidi, On Bank-Laine sequences, Complex Variables Theory Appl. 38 (1999), no. 3, 201-220. MR 1694317
  • [7] Janne Gröhn and Janne Heittokangas, New findings on the Bank-Sauer approach in oscillation theory, Constr. Approx. 35 (2012), no. 3, 345-361. MR 2914364, https://doi.org/10.1007/s00365-011-9137-8
  • [8] Janne Gröhn, Artur Nicolau, and Jouni Rättyä, Mean growth and geometric zero distribution of solutions of linear differential equations, to appear in J. Anal. Math. http://arxiv.org/abs/1410.2777
  • [9] Janne Gröhn and Jouni Rättyä, On oscillation of solutions of linear differential equations, J. Geom. Anal., DOI 10.1007/s12220-016-9701-3
  • [10] Einar Hille, Remarks on a paper be Zeev Nehari, Bull. Amer. Math. Soc. 55 (1949), 552-553. MR 0030000
  • [11] Einar Hille, Ordinary differential equations in the complex domain, Dover Publications, Inc., Mineola, NY, 1997. Reprint of the 1976 original. MR 1452105
  • [12] Ilpo Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1993. MR 1207139
  • [13] Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407
  • [14] Peter Lappan, A non-normal locally uniformly univalent function, Bull. London Math. Soc. 5 (1973), 291-294. MR 0330467
  • [15] Peter Lappan, The spherical derivative and normal functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 3 (1977), no. 2, 301-310. MR 509088, https://doi.org/10.5186/aasfm.1977.0312
  • [16] Peter Lappan, On the normality of derivatives of functions. II, J. London Math. Soc. (2) 24 (1981), no. 3, 495-501. MR 635880, https://doi.org/10.1112/jlms/s2-24.3.495
  • [17] Olli Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47-65. MR 0087746
  • [18] Zeev Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551. MR 0029999
  • [19] Christian Pommerenke, Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. MR 0507768
  • [20] Binyamin Schwarz, Complex nonoscillation theorems and criteria of univalence, Trans. Amer. Math. Soc. 80 (1955), 159-186. MR 0073038
  • [21] Norbert Steinmetz, Locally univalent functions in the unit disk, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 2, 325-332. MR 731787, https://doi.org/10.5186/aasfm.1983.0822
  • [22] Norbert Steinmetz, Normal families and linear differential equations, J. Anal. Math. 117 (2012), 129-132. MR 2944093, https://doi.org/10.1007/s11854-012-0017-3
  • [23] Shinji Yamashita, On normal meromorphic functions, Math. Z. 141 (1975), 139-145. MR 0379851

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Additional Information

Janne Gröhn
Affiliation: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
Email: janne.grohn@uef.fi

DOI: https://doi.org/10.1090/proc/13292
Keywords: Linear differential equation, normal function, oscillation theory, prescribed zeros, separation of zeros
Received by editor(s): February 2, 2016
Received by editor(s) in revised form: May 13, 2016
Published electronically: September 8, 2016
Additional Notes: The author was supported in part by the Academy of Finland, projects #258125 and #286877.
Communicated by: Stephen Ramon Garcia
Article copyright: © Copyright 2016 American Mathematical Society

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