Bank-Laine functions with sparse zeros

Author:
J. K. Langley

Journal:
Proc. Amer. Math. Soc. **129** (2001), 1969-1978

MSC (2000):
Primary 30D35; Secondary 34M05, 34M10

Published electronically:
November 30, 2000

MathSciNet review:
1825904

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**Steven B. Bank and Ilpo Laine,*On the oscillation theory of 𝑓′′+𝐴𝑓=0 where 𝐴 is entire*, Trans. Amer. Math. Soc.**273**(1982), no. 1, 351–363. MR**664047**, 10.1090/S0002-9947-1982-0664047-6**2.**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****3.**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**, 10.1007/BF02564659**4.**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**, 10.1007/BF03322360**5.**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**, 10.1017/S0013091500026857**6.**S. M. Elzaidi,*On Bank-Laine sequences*, Complex Variables Theory Appl.**38**(1999), no. 3, 201–220. MR**1694317****7.**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****8.**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**, 10.1112/jlms/s2-37.121.88**9.**W. K. Hayman,*Meromorphic functions*, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR**0164038****10.**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**, 10.5186/aasfm.1992.1723**11.**Einar Hille,*Ordinary differential equations in the complex domain*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. Pure and Applied Mathematics. MR**0499382****12.**Ilpo Laine,*Nevanlinna theory and complex differential equations*, de Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1993. MR**1207139****13.**J. K. Langley,*On second order linear differential polynomials*, Results Math.**26**(1994), no. 1-2, 51–82. MR**1290681**, 10.1007/BF03322288**14.**J. K. Langley,*Quasiconformal modifications and Bank-Laine functions*, Arch. Math. (Basel)**71**(1998), no. 3, 233–239. MR**1637382**, 10.1007/s000130050258**15.**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****16.**John Rossi,*Second order differential equations with transcendental coefficients*, Proc. Amer. Math. Soc.**97**(1986), no. 1, 61–66. MR**831388**, 10.1090/S0002-9939-1986-0831388-8**17.**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****18.**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**, 10.1090/S0002-9939-1985-0810160-8

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
30D35,
34M05,
34M10

Retrieve articles in all journals with MSC (2000): 30D35, 34M05, 34M10

Additional Information

**J. K. Langley**

Affiliation:
School of Mathematical Sciences, University of Nottingham, NG7 2RD United Kingdom

Email:
jkl@maths.nott.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-00-05779-8

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