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Bank-Laine functions with sparse zeros
Author(s):
J.
K.
Langley
Journal:
Proc. Amer. Math. Soc.
129
(2001),
1969-1978.
MSC (2000):
Primary 30D35;
Secondary 34M05, 34M10
Posted:
November 30, 2000
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Abstract:
A Bank-Laine function is an entire function  satisfying  at every zero of  . We construct a Bank-Laine function of finite order with arbitrarily sparse zero-sequence. On the other hand, we show that a real sequence of at most order 1, convergence class, cannot be the zero-sequence of a Bank-Laine function of finite order.
References:
- 1.
- S. Bank and I. Laine, On the oscillation theory of
 where  is entire, Trans. Amer. Math. Soc. 273 (1982), 351-363.MR 83k:34009 - 2.
- S. Bank and I. Laine, Representations of solutions of periodic second order linear differential equations, J. reine angew. Math. 344 (1983), 1-21.MR 85a:34008
- 3.
- -, On the zeros of meromorphic solutions of second-order linear differential equations, Comment. Math. Helv. 58 (1983), 656-677.MR 86a:34008
- 4.
- S. Bank, I. Laine and J. K. Langley, On the frequency of zeros of solutions of second order linear differential equations, Results. Math. 10 (1986), 8-24.MR 88c:34041
- 5.
- S. Bank and J. K. Langley, On the oscillation of solutions of certain linear differential equations in the complex domain, Proc. Edin. Math. Soc. 30 (1987), 455-469.MR 88i:30045
- 6.
- S. M. ElZaidi, On Bank-Laine sequences, Complex Variables 38 (1999), 201-200.MR 2000a:34170
- 7.
- J. B. Garnett, Bounded analytic functions, Academic Press, New York 1981.MR 83g:30037
- 8.
- G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), 88-104.MR 88m:30076
- 9.
- W. K. Hayman, Meromorphic functions, Oxford at the Clarendon Press, 1964. MR 29:1337
- 10.
- S. Hellerstein, J. Miles and J. Rossi, On the growth of solutions of certain linear differential equations, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 17 (1992), 343-365.MR 93m:34004
- 11.
- E. Hille, Ordinary differential equations in the complex domain, Wiley, New York, 1976.MR 58:17266
- 12.
- I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Math. 15, Walter de Gruyter, Berlin/New York 1993.MR 94d:34008
- 13.
- J. K. Langley, On second order linear differential polynomials, Results. Math. 26 (1994), 51-82.MR 95k:30059
- 14.
- -, Quasiconformal modifications and Bank-Laine functions, Arch. Math. 71 (1998), 233-239.MR 99e:34004
- 15.
- J. Miles and J. Rossi, Linear combinations of logarithmic derivatives of entire functions with applications to differential equations, Pacific J. Math. 174 (1996), 195-214.MR 97e:30055
- 16.
- J. Rossi, Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc. 97 (1986), 61-66.MR 87f:30078
- 17.
- L. C. Shen, Solution to a problem of S. Bank regarding the exponent of convergence of the solutions of a differential equation
 , Kexue Tongbao 30 (1985), 1581-1585.MR 87j:34020 - 18.
- -, Construction of a differential equation
 with solutions having prescribed zeros, Proc. Amer. Math. Soc. 95 (1985), 544-546.MR 87b:34005
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Additional Information:
J.
K.
Langley
Affiliation:
School of Mathematical Sciences, University of Nottingham, NG7 2RD United Kingdom
Email:
jkl@maths.nott.ac.uk
DOI:
10.1090/S0002-9939-00-05779-8
PII:
S 0002-9939(00)05779-8
Received by editor(s):
July 6, 1999
Received by editor(s) in revised form:
October 13, 1999
Posted:
November 30, 2000
Communicated by:
Albert Baernstein II
Copyright of article:
Copyright
2000,
American Mathematical Society
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