Electron. J. Diff. Equ., Vol. 2012 (2012), No. 107, pp. 1-8.

Sums of zeros of solutions to second order ODE with non-polynomial coefficients

Michael I. Gil'

Abstract:
We consider the equation $y''=F(z)y$ ( $z\in\mathbb{C}$) with an entire function $F$ satisfying the condition
$$ |F(z)|\le A \exp \big(\frac{|z|^\rho}{\rho}\big)\quad (\rho\ge 1,\; A=hbox{const}>0). $$
Let $z_k(y)$, $k=1, 2, \dots $ be the zeros of a solution $y(z)$ to the above equation. Bounds for the sums
$$ \sum_{k=1}^j \frac{1}{|z_k(y)|} \quad (j=1, 2, \dots) $$
are established. Some applications of these bounds are also considered.

Submitted September 8, 2011. Published June 25, 2012.
Math Subject Classifications: 34C10, 34A30.
Key Words: Complex differential equation; zeros of solutions.

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Michael I. Gil'
Department of Mathematics
Ben Gurion University of the Negev
P.0. Box 653, Beer-Sheva 84105, Israel
email: gilmi@bezeqint.net

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