On subharmonic and entire functions of small order: after Kjellberg

TL;DR

This paper introduces a new method for constructing transcendental entire functions of small order with controlled growth properties, using advanced harmonic function estimates in multiply connected domains.

Contribution

It develops a novel technique to bound the growth of positive harmonic functions, enabling precise construction of entire functions with specified size and shape characteristics.

Findings

Provides a general construction method for small order entire functions.

Establishes sharp growth estimates for harmonic functions in complex domains.

Enables control over the size of the set where the minimum modulus is large.

Abstract

We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves developing a new technique to give an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, giving a sharp estimate for the growth in many cases.