Hyperbolic Extension of Parabolic Trigonometry: Wilker, Lazarević, Wu-Debnah, Cusa-Huygens and Shafer Type Inequalities

Recently, a new generalization of hyperbolic functions called para-hyperbolic functions has been introduced. However, properties of the para-hyperbolic functions have not been investigated yet. In this paper, we derive the correct explicit formulas of the para-hyperbolic sine and cosine, study elementary properties of these functions, and explore which of the inequalities that hold for trigonometric and hyperbolic functions find their counterparts for para-hyperbolic functions. Namely, we prove a Wilker type inequality, Cusa-Huygens and Lazarević type inequality, Wu-Debnah modification of Wilker type inequality and Shafer type inequality for para-hyperbolic functions. Our results provide the first thorough exploration of para-hyperbolic functions which establishes groundwork for further discoveries and applications.