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Growth of solutions of an \(n\)-th order linear differential equation with entire coefficients. (English) Zbl 1037.34082
The authors discuss the growth of the solutions to higher-order linear differential equations of the form \[ f^{(n)}+ A_{n-1}(z) f^{(n-1)} + \cdots + A_0 (z)f =0, \] where \(A_0(z), \dots, A_{n-1}(z)\) are entire functions. They show that, if there exist a positive number \(\mu\) and a sequence \((z_j)_{j \in \mathbb{N}}\) with \(\lim _{j \to \infty} z_j = \infty\), and two real numbers \(\alpha, \beta\) \((0 \leq \beta < \alpha)\) such that \(| A_0(z_j)| \geq e^{\alpha | z_j| ^{\mu}}\) and \(| A_k(z_j)| \leq e^{\beta | z_j| ^{\mu}}\) as \(j \to + \infty\), \(k=1, \dots n-1\), then all solutions except zero are of infinite order of growth. This is a generalization of a result by G. G. Gundersen [Trans. Am. Math. Soc. 305, 415–429 (1988; Zbl 0669.34010)] for the second order case.

MSC:
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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