A necessary condition for all the zeros of an entire function of exponential type to lie in a curvilinear half-plane

Under the assumption that the integral
$$ \int_{\mathbb R}\frac{\log|F(x)|}{1+x^2}\,dx $$
exists, a condition necessary for all the zeros of the entire function $F(z)$ of exponential type to lie in the curvilinear half-plane $\operatorname{Im}z\leqslant\ (\geqslant)\ h(|\operatorname{Re}z|)$ (where $h(t)$ is a regularly varying function) is obtained.