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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
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