On Wilker’s and Huygen’s Type Inequalities for Generalized Trigonometric and Hyperbolic Functions

Abstract

Objectives: The Trigonometric inequalities, generalized trigonometric inequalities which have been obtained by Wilker and Cusa Huygens have attracted attention of so many researchers. Generalized trigonometric functions are simple generalization of the classical trigonometric functions. It is related to the r- Laplacian, which is known as a non-linear differential operator. Method: For the establishment of inequalities involving generalized trigonometric and hyperbolic functions convexity plays the important role in many aspects, also Monotonicity rule is used for sharpness of inequalities. This technique is used to refine and sharpness of inequalities. Findings: Our main result of this paper focus on generalization of Wilker and Cusa Huygens type inequalities for generalized trigonometric and hyperbolic functions with one parameter. Novelty: The inequalities with generalized trigonometric and hyperbolic functions proved in this research paper are Wilker's and Cusa Huygens generalization. It can be used for further refinement and sharpness. Keywords: Trigonometric function, Hyperbolic function, Generalized Trigonometric, Hyperbolic functions, Wilker Inequality and Huygen's Inequality