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The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc. (English) Zbl 1105.34059
The authors study the meromorphic solutions of a linear differential equation in the unit disc. They first give a definition of iterated order of a meromorphic function in the unit disc. They define the iterated $$n$$th-order $$\sigma_n(f)$$ of the meromorphic function $$f$$ as $\sigma_n(f)=\limsup_{r\to\infty}\frac{\log^{[n]}T(r,f)}{\log{\frac1{1-r}}},\quad n\in{\mathbb{N}},$ and the growth index of the iterated order of a meromorphic function $$f(z)$$ in the unit disc as $i(f)= \begin{cases} 0 &\text{if $$f$$ is nonadmissible}; \\ \min\{n: \sigma_n(f)<\infty\} &\text{if $$f$$ is admissible};\\ \infty &\text{if } \sigma_n(f)=\infty \text{ for all } n\in \mathbb{N}. \\ \end{cases}$ Then, they investigate the iterated order of analytic solutions of the following linear differential equation with analytic coefficients in the unit disc $f^{(k)}+a_{k-1}(z)f^{(k-1)}+\cdots +a_0(z)f=0.$ For example, they prove that if $$f$$ is an analytic solution of this equation, then $$i(f)\leq \max\{i(a_j)$$, $$j=1,\cdots, k-1\}.$$

##### MSC:
 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain 30D30 Meromorphic functions of one complex variable, general theory
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##### References:
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