Bank-Laine functions with sparse zeros
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- by J. K. Langley PDF
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Abstract:
A Bank-Laine function is an entire function $E$ satisfying $Eā(z) = \pm 1$ at every zero of $E$. 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
- Steven B. Bank and Ilpo Laine, On the oscillation theory of $f^{\prime \prime }+Af=0$ where $A$ is entire, Trans. Amer. Math. Soc. 273 (1982), no.Ā 1, 351ā363. MR 664047, DOI 10.1090/S0002-9947-1982-0664047-6
- Steven B. Bank and Ilpo Laine, Representations of solutions of periodic second order linear differential equations, J. Reine Angew. Math. 344 (1983), 1ā21. MR 716244, DOI 10.1515/crll.1983.344.1
- Steven B. Bank and Ilpo Laine, On the zeros of meromorphic solutions and second-order linear differential equations, Comment. Math. Helv. 58 (1983), no.Ā 4, 656ā677. MR 728459, DOI 10.1007/BF02564659
- Steven B. Bank, Ilpo Laine, and J. K. Langley, On the frequency of zeros of solutions of second order linear differential equations, Results Math. 10 (1986), no.Ā 1-2, 8ā24. MR 869795, DOI 10.1007/BF03322360
- Steven B. Bank and J. K. Langley, On the oscillation of solutions of certain linear differential equations in the complex domain, Proc. Edinburgh Math. Soc. (2) 30 (1987), no.Ā 3, 455ā469. MR 908453, DOI 10.1017/S0013091500026857
- S. M. Elzaidi, On Bank-Laine sequences, Complex Variables Theory Appl. 38 (1999), no.Ā 3, 201ā220. MR 1694317, DOI 10.1080/17476939908815165
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- Gary G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), no.Ā 1, 88ā104. MR 921748, DOI 10.1112/jlms/s2-37.121.88
- W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
- Simon Hellerstein, Joseph Miles, and John Rossi, On the growth of solutions of certain linear differential equations, Ann. Acad. Sci. Fenn. Ser. A I Math. 17 (1992), no.Ā 2, 343ā365. MR 1190329, DOI 10.5186/aasfm.1992.1723
- Einar Hille, Ordinary differential equations in the complex domain, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. MR 0499382
- Ilpo Laine, Nevanlinna theory and complex differential equations, De Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1993. MR 1207139, DOI 10.1515/9783110863147
- J. K. Langley, On second order linear differential polynomials, Results Math. 26 (1994), no.Ā 1-2, 51ā82. MR 1290681, DOI 10.1007/BF03322288
- J. K. Langley, Quasiconformal modifications and Bank-Laine functions, Arch. Math. (Basel) 71 (1998), no.Ā 3, 233ā239. MR 1637382, DOI 10.1007/s000130050258
- J. Miles and J. Rossi, Linear combinations of logarithmic derivatives of entire functions with applications to differential equations, Pacific J. Math. 174 (1996), no.Ā 1, 195ā214. MR 1398375
- John Rossi, Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc. 97 (1986), no.Ā 1, 61ā66. MR 831388, DOI 10.1090/S0002-9939-1986-0831388-8
- Li-Chien Shen, Solution to a problem of S. Bank regarding exponent of convergence of zeros of the solutions of differential equation $f''+Af=0$, Kexue Tongbao (English Ed.) 30 (1985), no.Ā 12, 1579ā1585. MR 850643
- Li-Chien Shen, Construction of a differential equation $y''+Ay=0$ with solutions having the prescribed zeros, Proc. Amer. Math. Soc. 95 (1985), no.Ā 4, 544ā546. MR 810160, DOI 10.1090/S0002-9939-1985-0810160-8
Additional Information
- J. K. Langley
- Affiliation: School of Mathematical Sciences, University of Nottingham, NG7 2RD United Kingdom
- MR Author ID: 110110
- Email: jkl@maths.nott.ac.uk
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1969-1978
- MSC (2000): Primary 30D35; Secondary 34M05, 34M10
- DOI: https://doi.org/10.1090/S0002-9939-00-05779-8
- MathSciNet review: 1825904