Inequalities for the generalized trigonometric and hyperbolic functions

Abstract

Abstract. The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p−Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinović-Adamović inequality, Lazarević's inequality, Huygens-type inequalities, Wilker-type inequalities, and CuzaHuygens-type inequalities, are generalized to the case of generalized functions.

On the Inequalities for the Generalized Trigonometric Functions

“…Recently several inequalities for these families of functions have been obtained. We refer the interested reader to the following papers [3,5,[7][8][9]13] and to the references therein.…”

“…We omit further details. Inequalities which connect the first and the third members of (35) and (39) have been established in [7] in Theorems 3.6 and 3.8, respectively. Our next goal is to provide short proofs of the first Huygens and the first Wilker inequalities for the generalized trigonometric functions.…”

“…Inequality (43) is established in [7,Theorem 3.22]. It is included here for the sake of completeness.…”

“…It is known that they are eigenfunctions of the Dirichlet problem for the one-dimensional -Laplacian. For more details concerning a recent progress in this rapidly growing area of functions theory the interested reader is referred to [1][2][3][4][5][6][7][8][9][10][11].…”

Inequalities for the generalized trigonometric and hyperbolic functions with two parameters

“…Motivated by this work, many authors have studied the equalities and inequalities related to generalized trigonometric and hyperbolic functions in [5,7,12]. Recently, in [17], S. Takeuchi has investigated the (p, q)−trigonometric functions depending on two parameters and in which the case of p = q coincides with the p−function of Lindqvist, and for p = q = 2 they coincide with familiar elementary functions.…”