Bank-Laine functions with sparse zeros

Langley, J.

doi:10.1090/S0002-9939-00-05779-8

Bank-Laine functions with sparse zeros
HTML articles powered by AMS MathViewer

by J. K. Langley PDF

Proc. Amer. Math. Soc. 129 (2001), 1969-1978 Request permission

Abstract:

A Bank-Laine function is an entire function $E$ satisfying $E’(z) = \pm 1$ 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. 1, 351–363. 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), 1–21. 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. 4, 656–677. 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. 1-2, 8–24. 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. 3, 455–469. MR 908453, DOI 10.1017/S0013091500026857
S. M. Elzaidi, On Bank-Laine sequences, Complex Variables Theory Appl. 38 (1999), no. 3, 201–220. 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. 1, 88–104. 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. 2, 343–365. 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. 1-2, 51–82. MR 1290681, DOI 10.1007/BF03322288
J. K. Langley, Quasiconformal modifications and Bank-Laine functions, Arch. Math. (Basel) 71 (1998), no. 3, 233–239. 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. 1, 195–214. MR 1398375
John Rossi, Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc. 97 (1986), no. 1, 61–66. 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. 12, 1579–1585. 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. 4, 544–546. MR 810160, DOI 10.1090/S0002-9939-1985-0810160-8