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Inequalities for zeros of solutions to second order ODE with one singular point. (English) Zbl 1335.34140

Summary: We consider the equation \[ y''+ P(z)y'+ Q(z)y= 0\;(z\in\mathbb{C}), \] where \[ P(z)= \sum^{n_P}_{k=0} p_kz^{k-1}\quad\text{and}\quad Q(z)= \sum^{n_Q}_{k=0} q_k z^{k-2} \] with real coefficients \(p_k\), \(q_j\) (\(k=0,, n_P\); \(j=0,\dots, n_Q\); \(n_P,n_Q<\infty\)).
Let \(z_k(y)\), \(k= 1,2,\dots\) be the nontrivial zeros of a solution \(y(z)\) to that equation. Estimates for the sums \[ \sum^j_{k=1} {1\over |z_k(y)|}\;(j= 1,2,\dots) \] are derived. Applications of the obtained estimates to the counting function of the zeros of solutions are also discussed.

MSC:

34M03 Linear ordinary differential equations and systems in the complex domain
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
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