Meromorphic functions with three radially distributed values

TL;DR

This paper investigates transcendental meromorphic functions with zeros, 1-points, and poles on three rays, establishing existence conditions and a normal family analogue based on the rays' spacing.

Contribution

The paper proves that such functions exist only when the three rays are equally spaced and extends the result to a normal family context.

Findings

Existence of functions only when rays are equally spaced

Normal family analogue of the main result

Characterization of distribution of zeros, 1-points, and poles

Abstract

We consider transcendental meromorphic functions for which the zeros, 1-points and poles are distributed on three distinct rays. We show that such functions exist if and only if the rays are equally spaced. We also obtain a normal family analogue of this result.