A generalization of Marcinkiewich's theorem on integer characteristic functions of probability distributions

The result of the paper is the following one: Let the function $\varphi(z)$ be analitic and of finite order $\rho>0$ in the upper half-plane. Suppose the function $\varphi(z)$ has no zeros and satisfies the following condition: $|\varphi(z)|\leq\varphi(i\operatorname{Im}z)$. Than $\rho\leq3$.