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
Abstract 
References 
Similar Articles 
Additional Information
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.
 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 (83k:34009), http://dx.doi.org/10.1090/S00029947198206640476
 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
(85a:34008)
 3.
Steven
B. Bank and Ilpo
Laine, On the zeros of meromorphic solutions and secondorder
linear differential equations, Comment. Math. Helv.
58 (1983), no. 4, 656–677. MR 728459
(86a:34008), http://dx.doi.org/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. 12, 8–24. MR 869795
(88c:34041), http://dx.doi.org/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
(88i:30045), http://dx.doi.org/10.1017/S0013091500026857
 6.
S.
M. Elzaidi, On BankLaine sequences, Complex Variables Theory
Appl. 38 (1999), no. 3, 201–220. MR 1694317
(2000a:34170)
 7.
John
B. Garnett, Bounded analytic functions, Pure and Applied
Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich,
Publishers], New YorkLondon, 1981. MR 628971
(83g:30037)
 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
(88m:30076), http://dx.doi.org/10.1112/jlms/s237.121.88
 9.
W.
K. Hayman, Meromorphic functions, Oxford Mathematical
Monographs, Clarendon Press, Oxford, 1964. MR 0164038
(29 #1337)
 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
(93m:34004)
 11.
Einar
Hille, Ordinary differential equations in the complex domain,
WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1976.
Pure and Applied Mathematics. MR 0499382
(58 #17266)
 12.
Ilpo
Laine, Nevanlinna theory and complex differential equations,
de Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter &
Co., Berlin, 1993. MR 1207139
(94d:34008)
 13.
J.
K. Langley, On second order linear differential polynomials,
Results Math. 26 (1994), no. 12, 51–82. MR 1290681
(95k:30059), http://dx.doi.org/10.1007/BF03322288
 14.
J.
K. Langley, Quasiconformal modifications and BankLaine
functions, Arch. Math. (Basel) 71 (1998), no. 3,
233–239. MR 1637382
(99e:34004), http://dx.doi.org/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
(97e:30055)
 16.
John
Rossi, Second order differential equations
with transcendental coefficients, Proc. Amer.
Math. Soc. 97 (1986), no. 1, 61–66. MR 831388
(87f:30078), http://dx.doi.org/10.1090/S00029939198608313888
 17.
LiChien
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
(87j:34020)
 18.
LiChien
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 (87b:34005), http://dx.doi.org/10.1090/S00029939198508101608
 1.
 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
 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), 88104.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), 343365.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), 5182.MR 95k:30059
 14.
 , Quasiconformal modifications and BankLaine functions, Arch. Math. 71 (1998), 233239.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), 195214.MR 97e:30055
 16.
 J. Rossi, Second order differential equations with transcendental coefficients, Proc. Amer. Math. Soc. 97 (1986), 6166.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), 15811585.MR 87j:34020
 18.
 , Construction of a differential equation with solutions having prescribed zeros, Proc. Amer. Math. Soc. 95 (1985), 544546.MR 87b:34005
Similar Articles
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:
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
