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On the growth of solutions of a class of higher order differential equations. (English) Zbl 1056.30029
Let $$f(z)$$ be a meromorphic function in the complex plane. Denote by $$\sigma(f)$$ the growth order of $$f(z)$$ and define a hyper-order of $$f(z)$$ by $\sigma_2(f)= \limsup_{r\to\infty} \log\log T(r,f)/\log r,$ where $$T(r, f)$$ is the characteristic function of $$f(z)$$. Let $$H_j(z)$$, $$j= 0,1,\dots,k- 1$$ be entire functions. The authors study linear differential equations of the form $f^{(k)}+ H_{k- 1} f^{(k- 1)}+\cdots+ H_s f^{(s)}+\cdots+ H_0f= 0.$ The “one dominate coefficient” case below is treated in this article. Let $$h_j(z)$$, $$j= 0,1,\dots,k -1$$ be entire functions with $$\sigma(h_j)< 1$$, and $$H_j(z)= h_j(z)e^{a_jz}$$, $$j= 0,1,\dots,k -1$$, where $$a_j$$, $$j= 0,1,\dots,k -1$$ are complex numbers. They suppose that there exist $$a_s$$ such that $$h_s(z)\not\equiv 0$$, and for $$j\neq s$$ if $$H_j(z)\not\equiv 0$$, $$a_j= c_j a_s$$, $$0< c_j< 1$$; if $$H_j(z)\equiv 0$$, define $$c_j= 0$$. Statements of their results are the following. Every transcendental solution of the differential equation above satisfies $$\sigma(f)=\infty$$. Further, if $$h_j(z)$$ are polynomials, then $$\sigma(f)= \infty$$ and $$\sigma_2(f)= 1$$. Main tools for the proofs are the Nevanlinna theory and the Wiman-Valiron theory. In particular, estimates for logarithmic derivatives of meromorphic functions due to G. G. Gundersen [J. Lond. Math. Soc., II. Ser. 37, No. 1, 88–104 (1988; Zbl 0638.30030)] play important roles.

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
##### Keywords:
complex oscillation; growing solutions; hyper-order
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