Applications of a duality between generalized trigonometric and hyperbolic functions

TL;DR

This paper explores a duality between generalized trigonometric and hyperbolic functions, enabling the transfer of properties and formulas between them, which simplifies deriving inequalities and multiple-angle formulas.

Contribution

It introduces a transformation linking generalized trigonometric and hyperbolic functions, allowing properties and formulas to be derived for both efficiently.

Findings

Established a transformation between the functions

Derived inequalities for both functions using the duality

Produced multiple-angle and double-angle formulas

Abstract

It is shown that generalized trigonometric functions and generalized hyperbolic functions can be transformed from each other. As an application of this transformation, a number of properties for one immediately lead to the corresponding properties for the other. In this way, Mitrinovi\'{c}-Adamovi\'{c}-type inequalities, multiple-angle formulas, and double-angle formulas for both can be produced.