A negative minimum modulus theorem and surjectivity of ultradifferential operators

TL;DR

This paper establishes the optimal conditions under which a minimum modulus theorem applies to certain entire functions, impacting the understanding of surjectivity of ultradifferential operators with constant coefficients.

Contribution

It proves the optimality of a classical minimum modulus theorem for canonical products and explores implications for ultradifferential operators' surjectivity.

Findings

The minimum modulus theorem is optimal under specified conditions.

Counterexamples exist when conditions are not met.

Results influence the theory of ultradifferential operators.

Abstract

In 1979 I. Cior\u{a}nescu and L. Zsid\'o have proved a minimum modulus theorem for entire functions dominated by the restriction to the positive half axis of a canonical product of genus zero, having all roots on the positive imaginary axis and satisfying a certain condition. Here we prove that the above result is optimal: if a canonical product {\omega} of genus zero, having all roots on the positive imaginary axis, does not satisfy the condition in the 1979 paper, then always there exists an entire function dominated by the restriction to the positive half axis of {\omega}, which does not satisfy the desired minimum modulus conclusion. This has relevant implication concerning the subjectivity of ultra differential operators with constant coefficients.