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

We consider the equation $y''=F(z)y$ ($zinmathbb{C}$) with an entire function $F$ satisfying the condition $$ |F(z)|le A exp ig(frac{|z|^ ho}{ ho}ig)quad ( hoge 1,; A= ext{const}>0). $$ 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.