On Complex Oscillation Theory of Solutions of Some Higher Order Linear Differential Equations
Received:November 27, 2011  Revised:March 27, 2012
Key Word: complex differential equations   entire function   the growth of order   the exponent of convergence of the zeros.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11171080) and Foundation of Science and Technology Department of Guizhou Province (Grant No.[2010] 07).
 Author Name Affiliation Jianren LONG School of Mathematics and Computer Science, Guizhou Normal University, Guizhou 550001, P. R. China Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China
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In this paper, we shall use Nevanlinna theory of meromorphic functions to investigate the complex oscillation theory of solutions of some higher order linear differential equation. Suppose that $A$ is a transcendental entire function with $\rho(A)<\frac{1}{2}$. Suppose that $k\geq 2$ and $f^{(k)}+A(z)f=0$ has a solution $f$ with $\lambda(f)<\rho(A)$, and suppose that $A_{1}=A+h$, where $h\not\equiv 0$ is an entire function with $\rho(h)<\rho(A)$. Then $g^{(k)}+A_{1}(z)g=0$ does not have a solution $g$ with $\lambda(g)<\infty$.