# On the Heins Theorem

## TL;DR

This paper investigates pairs of entire functions of finite exponential type with bounded minimum modulus, revealing they must be bounded on certain rotating half-planes and providing bounds for these regions.

## Contribution

It extends understanding of the behavior of entire functions of finite exponential type, specifically characterizing their boundedness on rotating half-planes.

## Key findings

- Functions must be bounded on some rotating half-planes
- Derived bounds for rotation functions of these half-planes
- Clarified limitations of Heins Theorem extension

## Abstract

It is known that the famous Heins Theorem (also known as the de Branges Lemma) about the minimum of two entire functions of minimal type does not extend to functions of finite exponential type. We study in detail pairs of entire functions $f, g$ of finite exponential type satisfying $\sup_{z\in\mathbb{C}}\min\{|f(z)|,|g(z)|\}<\infty.$ It turns out that $f$ and $g$ have to be bounded on some rotating half-planes. We also obtain very close upper and lower bounds for possible rotation functions of these half-planes.