BankLaine functions with sparse zeros
Author:
J. K. Langley
Journal:
Proc. Amer. Math. Soc. 129 (2001), 19691978
MSC (2000):
Primary 30D35; Secondary 34M05, 34M10
Published electronically:
November 30, 2000
MathSciNet review:
1825904
Fulltext PDF Free Access
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Abstract: A BankLaine function is an entire function satisfying at every zero of . We construct a BankLaine function of finite order with arbitrarily sparse zerosequence. On the other hand, we show that a real sequence of at most order 1, convergence class, cannot be the zerosequence of a BankLaine function of finite order.
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 S. Bank and I. Laine, On the oscillation theory of where is entire, Trans. Amer. Math. Soc. 273 (1982), 351363.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), 121.MR 85a:34008
 3.
 , On the zeros of meromorphic solutions of secondorder linear differential equations, Comment. Math. Helv. 58 (1983), 656677.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), 824.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), 455469.MR 88i:30045
 6.
 S. M. ElZaidi, On BankLaine sequences, Complex Variables 38 (1999), 201200.MR 2000a:34170
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 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), 88104.MR 88m:30076
 9.
 W. K. Hayman, Meromorphic functions, Oxford at the Clarendon Press, 1964. MR 29:1337
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 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), 343365.MR 93m:34004
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 E. Hille, Ordinary differential equations in the complex domain, Wiley, New York, 1976.MR 58:17266
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 I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Math. 15, Walter de Gruyter, Berlin/New York 1993.MR 94d:34008
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 J. K. Langley, On second order linear differential polynomials, Results. Math. 26 (1994), 5182.MR 95k:30059
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 , Quasiconformal modifications and BankLaine functions, Arch. Math. 71 (1998), 233239.MR 99e:34004
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 J. Miles and J. Rossi, Linear combinations of logarithmic derivatives of entire functions with applications to differential equations, Pacific J. Math. 174 (1996), 195214.MR 97e:30055
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 J. Rossi, Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc. 97 (1986), 6166.MR 87f:30078
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 , Construction of a differential equation with solutions having prescribed zeros, Proc. Amer. Math. Soc. 95 (1985), 544546.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:
http://dx.doi.org/10.1090/S0002993900057798
PII:
S 00029939(00)057798
Received by editor(s):
July 6, 1999
Received by editor(s) in revised form:
October 13, 1999
Published electronically:
November 30, 2000
Communicated by:
Albert Baernstein II
Article copyright:
© Copyright 2000 American Mathematical Society
