The growth of linear differential equations and their applications.

*(English)*Zbl 1131.34059This 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 \(\sigma_1(f)\).

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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##### References:

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