Sufficient sets in a certain space of entire functions

For any trigonometrically convex function $h(\varphi)$ an entire function $L(z)$ is constructed, satisfying the relation
$$ \ln|L(re^{i\varphi})|=h(\varphi)r+O(r^{1/2}\ln r),\qquad re^{i\varphi}\notin\Omega(a_n), $$
where the $a_n$ are the zeros of $L(z)$ and $\Omega(a_n)=\{z:|z-a_n|\leqslant1\}$. The set of zeros of such a function is sufficient in the space of entire functions $F(z)$ satisfying
$$ \sup_{r,\varphi}\frac{\ln|F(re^{i\varphi})|}{h(\varphi)r-r^{q+\varepsilon}}<\infty $$
for some $\varepsilon>0$, where $q\in(1/2,1)$ is a parameter of the space.
Bibliography: 5 titles.