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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 $$\rho_2(f)\geq\mu$$.
(2) The standard notation for the hyperorder is currently $$\sigma_2(f)$$ (or $$p_2(f))$$ instead of $$\sigma_1(f)$$.