On the least type of entire functions of order $\rho\in(0,1)$ with positive zeros

The paper is devoted to the theory of extremal problems in classes of entire functions with constraints on the growth and distribution of zeros and is associated with problems of completeness of exponential systems in the complex domain. The question of finding the exact lower bound for types of all entire functions of order $\rho\in(0,1)$ whose zeros lie on the ray and have prescribed upper $\rho$-density and $\rho$-step is discussed. It is shown that the infimum is attained in this problem, and a detailed construction of the extremal function is given. This result gives a complete solution of the extremal problem and generalizes preceding result of A. Yu. Popov.