Analytic functions of infinite order in half-plane

J. B. Meles (1979) considered entire functions with zeros restricted to a finite number of rays. In particular, it was proved that if $f$ is an entire function of infinite order with zeros restricted to a finite number of rays, then its lower order equals infinity. In this paper, we prove a similar result for a class of functions analytic in the upper half-plane. The analytic function $f$ in $\mathbb{C}_+=\{z:\Im z>0\}$ is called proper analytic if $\limsup\limits_{z\to t}\ln|f(z)|\leq 0$ for all real numbers $t\in\mathbb{R}$. The class of the proper analytic functions is denoted by $JA$. The full measure of a function $f\in JA$ is a positive measure, which justifies the term "proper analytic function". In this paper, we prove that if a function $f$ is the proper analytic function in the half-plane $\mathbb{C}_+$ of infinite order with zeros restricted to a finite number of rays $\mathbb{L}_k$ through the origin, then its lower order equals infinity.