ANALYTIC FUNCTIONS OF INFINITE ORDER IN HALF-PLANE

Abstract

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 C+ = {𝑧 : Im 𝑧 > 0} is called proper analytic if lim sup(z→t)⁡( ln⁡|f(z)|)≤0 for all real numbers 𝑡 ∈ R. The class of the proper analytic functions is denoted by 𝐽𝐴. The full measure of a function 𝑓 ∈ 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 C+ of infinite order with zeros restricted to a finite number of rays L𝑘 through the origin, then its lower order equals infinity.

Keywords

• half-plane
• proper analytic function
• infinite order
• lower order
• fourier coefficients
• full measure