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Oscillation of fast growing solutions of linear differential equations in the unit disc. (English) Zbl 1215.34112
Consider the homogeneous differential equation
\[ f^{(k)} +A(z) f=0,\quad k\geq 2, \]
where \(A(z)\not\equiv 0\) is an analytic function in the unit disc of finite iterated \(p\)-order \(\sigma\). The author proves that, given an analytic function \(\varphi\) of finite iterated \(p\)-order such that \(\varphi^{(j)}(z)\not\equiv 0\), \(j\in \{0, \dots, k\}\), for any nontrivial solution \(f\), the counting function of zeros of \(f^{(j)}(z)-\varphi(z)\) has \((p+1)\)-order \(\sigma\), \(j\in \{0, \dots, k\}\). This is an elementary generalization of the known result for the case \(\varphi\equiv 0\).

34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34M03 Linear ordinary differential equations and systems in the complex domain
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