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Growth of solutions of second order linear differential equations


Authors: Ilpo Laine and Pengcheng Wu
Journal: Proc. Amer. Math. Soc. 128 (2000), 2693-2703
MSC (1991): Primary 34A20; Secondary 30D05, 30D20, 30D35
DOI: https://doi.org/10.1090/S0002-9939-00-05350-8
Published electronically: March 1, 2000
MathSciNet review: 1664418
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Abstract: We consider the differential equation $f''+A(z)f'+ B(z)f=0$, where $A(z)$ and $B(z)$ are entire functions. Provided $\rho (B)<\rho (A)$ and $T(r,A)\sim \log M(r,A) $ as $r\to \infty $outside a set of finite logarithmic measure, we prove that all nonconstant solutions $f$ of this equation are of infinite order.


References [Enhancements On Off] (What's this?)

  • [1] A. Edrei and W.H.J. Fuchs, The deficiencies of meromorphic functions of order less than one, Duke Math. J. 27 (1960), 233-249. MR 23:A1039
  • [2] G.G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. 37 (1988), 88-104. MR 88m:30076
  • [3] G.G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), 415-429. MR 88j:34016
  • [4] W.K. Hayman, On Iversen's theorem for meromorphic functions with few poles, Acta Math. 141 (1978), 115-145. MR 58:11409
  • [5] W.K. Hayman, Meromorphic functions, Oxford University Press, Oxford, 1964. MR 29:1337
  • [6] W.K. Hayman and J.F. Rossi, Characteristic, maximum modulus and value distribution, Trans. Amer. Math. Soc. 284 (1984), 651-664.
  • [7] W.K. Hayman and F.M. Stewart, Real inequalities with application to function theory, Proc. Cambridge Philos. Soc. 50 (1953), 250-260. MR 15:857g
  • [8] S. Hellerstein, J. Miles and J. Rossi, On the growth of solutions of $f''+gf'+hf=0$, Trans. Amer. Math. Soc. 324 (1991), 693-706. MR 91h:30047
  • [9] T. Murai, The deficiency of entire functions with Fejér gaps, Ann. Inst. Fourier (Grenoble) 33 (1983), 39-58. MR 84m:30046
  • [10] H. Wittich, Zur Theorie linearer Differentialgleichungen im Komplexen, Ann. Acad. Sci. Fenn. Ser. A I 379 (1966), 12. MR 33:5989
  • [11] G.H. Zhang, Theory of entire and meromorphic functions, deficient and asymptotic values and singular directions, Providence, Rhode Island, 1993. MR 94h:30039

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Additional Information

Ilpo Laine
Affiliation: Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland
Email: Ilpo.Laine@joensuu.fi

Pengcheng Wu
Affiliation: Department of Mathematics, Nationality Institute of Guizhou, Guiyang, 550025, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-00-05350-8
Keywords: Differential equation, entire function, finite order, Nevanlinna characteristic
Received by editor(s): August 18, 1998
Received by editor(s) in revised form: October 27, 1998
Published electronically: March 1, 2000
Additional Notes: This work was supported in part by the Finnish Academy research grant 37701.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society

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