Sunyer-i-Balaguer's Almost Elliptic Functions and Yosida's Normal Functions

TL;DR

This paper explores two classes of meromorphic functions, almost elliptic and Yosida's normal functions, providing conditions for their zeros and poles, and constructing explicit examples of these functions.

Contribution

It offers the first explicit examples of almost elliptic functions and characterizes Yosida's normal functions of the first category through zeros and poles.

Findings

Provided sufficient conditions for zeros and poles of almost elliptic functions.

Established necessary and sufficient conditions for Yosida's normal functions.

Derived a parametric representation for Yosida's normal functions.

Abstract

We study the properties of two classes of meromorphic functions in the complex plane. The first one is the class of almost elliptic functions in the sense of Sunyer-i-Balaguer. This is the class of meromorphic functions f such that the family of shifts f(z+h) (h are complex numbers) is normal with respect to the uniform convergence in the whole complex plane. Given two sequences of complex numbers, we provide sufficient conditions for them to be zeros and poles of some almost elliptic function. These conditions enable one to give (for the first time) explicit non-trivial examples of almost elliptic functions. The second class was introduced by K.Yosida, who called it a class of normal functions of the first category. This is the class of meromorphic functions f such that the family of shifts f(z+h)is normal with respect to the uniform convergence on compacta in the complex plane and no limit point of the family is a constant function. We give necessary and sufficient conditions for two sequences of complex numbers to be zeros and poles of some normal function of the first category and obtain a parametric representation for this class in terms of zeros and poles.