Inequalities for the Jacobian elliptic functions with complex modulus

TL;DR

This paper establishes new inequalities for the absolute values of Jacobian elliptic functions when their complex modulus is outside the usual unit interval, expanding understanding of these functions in complex analysis.

Contribution

It introduces novel inequalities for Jacobian elliptic functions with complex moduli beyond the unit interval, filling a gap in existing mathematical literature.

Findings

Proved inequalities for elliptic functions with complex modulus

Extended the theory of Jacobian elliptic functions beyond real moduli

Provided tools for further research in complex elliptic functions

Abstract

Despite the fact that there is a huge amount on papers and books devoted to the theory of Jacobian elliptic functions, very little is known when the modulus $k$ of these functions lies outside the unit interval $[0,1]$. In this note, we prove some simple inequalities for the absolute value of Jacobian elliptic functions with complex modulus.