Growth of solutions of second order linear differential equations

Laine, Ilpo; Wu, Pengcheng

doi:10.1090/S0002-9939-00-05350-8

Growth of solutions of second order linear differential equations
HTML articles powered by AMS MathViewer

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