Applications of a duality between generalized trigonometric and hyperbolic functions II

TL;DR

This paper explores the duality between generalized trigonometric and hyperbolic functions, applying it to establish various inequalities and derive new multiple- and double-angle formulas.

Contribution

It introduces new inequalities and multiple- and double-angle formulas for generalized functions using the duality framework.

Findings

Proved dual pairs of inequalities for generalized functions.

Derived new multiple- and double-angle formulas.

Enhanced understanding of the duality between generalized functions.

Abstract

Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of Wilker-type inequalities, Huygens-type inequalities, and (relaxed) Cusa-Huygens-type inequalities for the generalized functions. In addition, multiple- and double-angle formulas not previously obtained are also given.