Wilker–Huygens type inequalities for  p -generalized lemniscate and hyperbolic lemniscate functions

Abstract

This paper extends Wilker–Huygens inequalities to p -generalized (arc) lemniscate functions. First, we define p -generalized lemniscate functions and their hyperbolic analogues based on the ( p , 2 p )-generalized trigonometric and hyperbolic functions. Then, we establish key monotonicity lemmas for auxiliary functions by means of the L’Hôpital Monotone Rule and derivative calculations. Finally, we prove a series of Wilker–Huygens type inequalities for p -generalized (arc) lemniscate functions and their hyperbolic versions, and derive corresponding corollaries. When p = 2, these results reduce to the classical inequalities for lemniscate functions.