Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 107, pp. 1-8.

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

Author: Michael I. Gil (Ben Gurion Univ. of the Negev, Israel)

Abstract: 
 We consider the equation $y''=F(z)y$ ($z\in\mathbb{C}$)
 with an entire function $F$  satisfying the condition
 $$
 |F(z)|\le A \exp \big(\frac{|z|^\rho}{\rho}\big)\quad (\rho\ge 1,\; A=\text{const}>0).
 $$
 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.