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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 \(i(f)\leq \max\{i(a_j)\), \(j=1,\cdots, k-1\}.\)

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