Growth of solutions of second order linear differential equations
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- by Ilpo Laine and Pengcheng Wu PDF
- Proc. Amer. Math. Soc. 128 (2000), 2693-2703 Request permission
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
- Albert Edrei and Wolfgang H. J. Fuchs, The deficiencies of meromorphic functions of order less than one, Duke Math. J. 27 (1960), 233–249. MR 123717
- 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
- Wei Xin Gao, Linear differential equations with regular singularities, Acta Math. Sinica 30 (1987), no. 4, 566–576 (Chinese). MR 917119
- W. K. Hayman, On Iversen’s theorem for meromorphic functions with few poles, Acta Math. 141 (1978), no. 1-2, 115–145. MR 492265, DOI 10.1007/BF02545745
- W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
- W.K. Hayman and J.F. Rossi, Characteristic, maximum modulus and value distribution, Trans. Amer. Math. Soc. 284 (1984), 651–664.
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- Simon Hellerstein, Joseph Miles, and John Rossi, On the growth of solutions of $f''+gf’+hf=0$, Trans. Amer. Math. Soc. 324 (1991), no. 2, 693–706. MR 1005080, DOI 10.1090/S0002-9947-1991-1005080-X
- Takafumi Murai, The deficiency of entire functions with Fejér gaps, Ann. Inst. Fourier (Grenoble) 33 (1983), no. 3, 39–58 (English, with French summary). MR 723947
- H. Wittich, Zur Theorie linearer Differentialgleichungen im Komplexen, Ann. Acad. Sci. Fenn. Ser. A I No. 379 (1966), 19 (German). MR 0197828
- Guan Hou Zhang, Theory of entire and meromorphic functions, Translations of Mathematical Monographs, vol. 122, American Mathematical Society, Providence, RI, 1993. Deficient and asymptotic values and singular directions; Translated from the Chinese by Chung-chun Yang. MR 1222870, DOI 10.1126/science.262.5132.374
Additional Information
- Ilpo Laine
- Affiliation: Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland
- MR Author ID: 109330
- Email: Ilpo.Laine@joensuu.fi
- Pengcheng Wu
- Affiliation: Department of Mathematics, Nationality Institute of Guizhou, Guiyang, 550025, People’s Republic of China
- 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
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1664418