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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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