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Proof of a conjecture of Bank and Laine regarding the product of two linearly independent solutions of $ y''+Ay=0$


Author: Li-Chien Shen
Journal: Proc. Amer. Math. Soc. 100 (1987), 301-308
MSC: Primary 34A20; Secondary 30D35
DOI: https://doi.org/10.1090/S0002-9939-1987-0884470-4
MathSciNet review: 884470
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Abstract: Let $ A$ be a transcendental entire function of order $ < 1$. If $ {w_1}$ and $ {w_2}$ are two linearly independent solutions of the differential equation $ y'' + Ay = 0$, then at least one of $ {w_1},{w_2}$ has the property that the exponent of convergence of its zeros is $ > 1$.


References [Enhancements On Off] (What's this?)

  • [1] K. Arima, On maximum modulus of integral functions, J. Math. Soc. Japan 4 (1952), 62-66. MR 0049320 (14:155d)
  • [2] S. Bank and I. Laine, On the oscillation theory of $ f'' + Af = 0$ where $ A$ is entire, Trans. Amer. Math. Soc. 273 (1982), 351-363. MR 664047 (83k:34009)
  • [3] -, On the zeros of meromorphic solutions of second order linear differential equations, Comment. Math. Helv. 58 (1983). MR 728459 (86a:34008)
  • [4] A. Edrei, The problem of Bank and Laine, unpublished manuscript.
  • [5] B. Ja. Levin, Distribution of zeros of entire functions, Transl. Math. Monos., vol. 5, Amer. Math. Soc., Providence, R. I., 1963. MR 589888 (81k:30011)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0884470-4
Keywords: Linearly independent solutions, entire functions, Cartan's lemma, Carleman's differential inequality
Article copyright: © Copyright 1987 American Mathematical Society

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