Inequalities for the generalized trigonometric and hyperbolic functions

Abstract In this paper, the authors present some inequalities of the generalized trigonometric and hyperbolic functions which occur in the solutions of some linear differential equations and physics. By these results, some well-known classical inequalities for them are improved, such as Wilker inequality, Huygens inequality, Lazarević inequality and Cusa-Huygens inequality.


Introduction
The well-known Wilker inequality for trigonometric functions x π sin tan 2, for 0, 2 2 (1) was proposed by Wilker [1] and proved by Sumner et al. [2]. The hyperbolic counterpart of (1) was established in [3] as follows: x sinh tanh 2, for 0, . 2 (2) A related inequality that is of interest to us is the Huygens inequality [4,5]: x x x π 2 sin tan 3 , for 0, 2 , The Wilker inequalities (1), (2) and the Huygens inequalities (3), (4) have attracted much interest of many mathematicians. Many generalizations, improvements and refinements of the Wilker inequality and the Huygens inequality can be found in the literature [6,7] and references therein.
In [6,Theorems 5 and 8] tanh 3 , for 0, and 1 2, In recent years, the following two-sided trigonometric inequality for hyperbolic functions , for 0, 2 1 3 has attracted attention of several research studies. The left inequality of (9) is called the Lazarević inequality, which is obtained in [8]. The right inequality of (9) is the famous Cusa-Huygens inequality, which is obtained in [9], The counterpart of (9) for trigonometric functions is also well known. The left inequality (10) has been proven by Mitrinović [10], while the second one by Cusa and Huygens [4,11]. The aforementioned inequalities have also been obtained in [5].
The generalized trigonometric and hyperbolic functions depending on a parameter > p 1 were studied by Lindqvist in a highly cited paper [12]. Drábek and Manásevich [13] considered a certain ( ) p q , -eigenvalue problem with the Dirichlét boundary condition and found the complete solution to the problem. The solution of a special case is the function sin p,q , which is the first example of the so-called ( ) p q , -trigonometric function. Motivated by the ( ) p q , -eigenvalue problem, Takeuchi [14] has investigated the ( ) p q ,trigonometric functions depending on two parameters 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. In [15], the relations of generalized trigonometric and hyperbolic functions of two parameters with their inverse functions were studied. In [16], some inequalities for ( ) p q , -trigonometric were obtained and a few conjectures for them were posed. Recently, a conjecture posed in [16] was verified in [17]. In [18], the power mean inequality for generalized trigonometric and hyperbolic functions with two parameters was presented.
Motivated by these results on the trigonometric functions, we make a contribution to the subject by showing some Wilker inequalities, Huygens inequalities, Lazarević inequalities and Cusa-Huygens inequalities for the ( ) p q , -trigonometric and hyperbolic functions.

Definitions and formulas
For the formulation of our main results, we give the following definitions of ( ) p q , -trigonometric and hyperbolic functions, such as the generalized ( ) p q , -cosine function, the generalized ( ) p q , -tangent function and their inverses, and also the corresponding hyperbolic functions. For is defined by Inequalities for the generalized trigonometric and hyperbolic functions  1581 is called the generalized ( ) p q , -sine function, denoted by sin : is defined as The generalized ( is defined as Similarly, for ∈ ( ∞) x 0, , the inverse of the generalized ( ) p q , -hyperbolic sine function [15] is defined by and also other corresponding ( ) p q , -hyperbolic functions, such as ( ) p q , -hyperbolic cosine and tangent functions, are defined by It is clear that all these generalized functions coincide with the classical ones when = = p q 2.

Preliminaries and proofs
In this section, we give three Lemmas needed in the proofs of our main results. First, let us recall the following well-known formulas [15,19]: for and the Jacobsthal inequality [20] The following Lemmas will be frequently applied later.
3 . In this case, by (14), (16), (17) and (28)  Proof. By differentiation, we have It is true by (18) and (19), which implies that h is strictly decreasing. Hence, the other conclusion for h is clear. □

Main results
Proof.