Miles problem for analytic functions of infinite order on the half-plane defined by a model function

J. B. Miles (1979) considered entire functions with zeros dis- tributed on a finite system of rays. In particular, it was proved that if f is an entire function of infinite order whose zeros are located on a fi- nite system of rays, then its lower order is also equal to infinity. K. G. Malyutin, M. V. Kabanko, and T. V. Shevtsova (2022) extended Miles’s result to truly analytic functions of infinite order with respect to the classi- cal growth function on the upper half-plane. An analytic function f on the upper half-plane of a complex variable is called truly analytic if its upper limit on the real axis is not positive. The total measure of a truly ana- lytic function is a positive measure, which justifies the term ”truly analytic function”. If the order of a truly analytic function is equal to infinity, then the function is called a function of infinite order. Otherwise, the function f is called a function of finite order. In this paper we prove a similar result in the space of functions of infinite order with respect to the model growth function of analytic functions on the upper half-plane.