# On the range of harmonic maps in the plane

## TL;DR

This paper generalizes classical results about harmonic functions, specifically the Little Picard Theorem and Liouville theorem, by examining the range of harmonic maps in the plane, leading to broader conditions for constancy.

## Contribution

It extends Lewis's harmonic proof of Picard's theorem and harmonic Liouville theorem to a more general setting involving the range of harmonic maps in the plane.

## Key findings

- Generalization of Lewis's theorem to broader harmonic maps
- New conditions under which harmonic maps are constant
- Extension of harmonic Liouville theorem

## Abstract

In 1994 J. Lewis obtained a purely harmonic proof of the classical Little Picard Theorem by showing that if the joint value distribution of two entire harmonic functions satisfies certain restrictions then they are necessarily constant. We generalize Lewis'theorem and the harmonic Liouville theorem in terms of the range of a harmonic map in the plane.