On a result of Hayman concerning the maximum modulus set

TL;DR

This paper determines the exact number of disjoint analytic curves forming the maximum modulus set of entire functions near zero, extending Hayman's 1951 results and proposing a conjecture for the general case.

Contribution

It establishes the precise count of these curves for all entire functions except a small algebraic set and introduces new structural insights near the origin.

Findings

Exact number of curves for most entire functions identified

Structural properties of the maximum modulus set near zero clarified

Conjecture proven for polynomials of degree less than four

Abstract

The set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the exact number of these curves for all entire functions, except for a "small" set whose Taylor series coefficients satisfy a certain simple, algebraic condition. Moreover, we give new results concerning the structure of this set near the origin, and make an interesting conjecture regarding the most general case. We prove this conjecture for polynomials of degree less than four.