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The growth of linear differential equations and their applications. (English) Zbl 1131.34059
This paper is devoted to considering growth properties of complex linear differential equations. Applications to some shared value problems are given as well. A typical result reads as follows: Let \(E\) be a set of complex numbers such that \(\overline{\roman{dens}}|E|>0\), and let \(A_0,\dots,A_n\) be entire functions such that \(|A_0(z)|\geq e^{\alpha|z|^\mu}\) and \(|A_j(z)|\leq e^{\beta|z|^\mu}\), \(j=1,\dots,n\), as \(z\to\infty\), \(z\in E\). Here \(0\leq\beta<\alpha\) and \(\mu>0\). Then every nontrivial meromorphic solution \(f\) of \(A_n(z)f^{(n)}+\dots+A_0(z)f=0\) is of infinite order and of hyperorder \(\rho_2(f)\geq\mu\).
We add the following remarks:
(1) Lemma 2 is already well known [see e.g. G. G. Gundersen, Trans. Am. Math. Soc. 305, No. 1, 415–429 (1988; Zbl 0669.34010)].
(2) The standard notation for the hyperorder is currently \(\sigma_2(f)\) (or \(p_2(f))\) instead of \(\sigma_1(f)\).

MSC:
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D10 Representations of entire functions of one complex variable by series and integrals
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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