讨论一类高阶亚纯系数非齐次线性微分方程解的零点问题,当方程的系数A0是亚纯函数且满足δ(∞,A0)=δ(>0)和limr→∞ logT(r,A0)/log r=∞时,如果f1和f2是方程fk + Ak-1fk-1+…+A0f=F的两个线性无关解,得到max{λ(f1),λ(f2)}=∞.还考虑了 sigma(F)=∞或Ad (1≤ d ≤k-1)满足limr→∞ logm(r,Ad)/log r=∞的情况.
Abstract
We investigate the problems of zeros of solutions for some higher order nonhomogeneous linear differential equations. When A0 is a meromorphic function that satisfies δ(∞,A0) = δ(> 0) and limr→∞ logT(r,A0)/log r=∞, if f1 and f2 are two linearly independent meromorphic solutions of equation fk + Ak-1fk-1+…+A0f=F,we obtain max{λ(f1),λ(f2)}=∞.And we also investigate the cases that σ(F) = ∞, or some Ad (1≤ d ≤k-1) satisfies limr→∞ logm(r,Ad)/log r=∞.
关键词
微分方程 /
零点收敛指数 /
增长级
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Key words
differential equation /
exponent of convergence of zeros /
order of growth
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参考文献
[1] Chen Z. X., On the complex oscillation theory of fk +Af = F, Proc. Edinburgh Mathematical Society, 1993, 36: 447-461.
[2] Chen Z. X., Zero of meromorphic solution of higher order linear differential equatians, Analysis, 1994, 14: 425-438.
[3] Chen Z. X., Gao S. A., On the complex oscillation of non-homogeneous linear differential equatians with polynomial coefficients, Kodai Math. J., 1992, 15: 65-78.
[4] Frank G., Hellerstein S., On the meromorphic solution of non-homogeneous linear differential equatians with polynomial coefficients, Proc. London Math. Soc., 1986, 53(3): 407-428.
[5] Gao S. A., On the complex oscillation of non-homogeneous linear differential equatians with polynomial coefficients, Comment. Math. Univ. Sancti Pauli, 1989, 38(1): 11-20.
[6] Hayman W., Meromorphic Functions, Clarendon Press, Oxford, 1964.
[7] Laine I., A note on the complex oscillation theory of non-homogeneous linear differential equatians, Result in Math., 1990, 18: 282-285.
[8] Yang L., Value Distribution Theory and New Research (in Chinese), Science Press, Beijing, 1982.
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脚注
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