On the Inequalities for the Generalized Trigonometric Functions

Recently the generalized trigonometric and the generalized hyperbolic functions have attracted attention of several researches. These functions, introduced by Lindqvist in [1], depend on one parameter p > 1. They become classical trigonometric and hyperbolic functions when p = 2. It is known that they are eigenfunctions of the Dirichlet problem for the one-dimensional p-Laplacian. For more details concerning a recent progress in this rapidly growing area of functions theory the interested reader is referred to [1–11]. The goal of this paper is to establish some inequalities for families of functions mentioned earlier in this section. In Section 2 we give definitions of functions under discussions. Also, some preliminary results are included there. Some useful inequalities utilized in this note are established in Section 3. The main results, involving the Huygens-type and the Wilker-type inequalities, are derived in Section 4.


Introduction
Recently the generalized trigonometric and the generalized hyperbolic functions have attracted attention of several researches.These functions, introduced by Lindqvist in [1], depend on one parameter  > 1.They become classical trigonometric and hyperbolic functions when  = 2.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].
The goal of this paper is to establish some inequalities for families of functions mentioned earlier in this section.In Section 2 we give definitions of functions under discussions.Also, some preliminary results are included there.Some useful inequalities utilized in this note are established in Section 3. The main results, involving the Huygens-type and the Wilker-type inequalities, are derived in Section 4.
In what follows, let the parameter  be strictly greater than 1.In some cases this assumption will be relaxed to 1 <  ≤ 2. We will adopt notation and definitions used in [5].Let Further, let ) .
A remarkable result states that the mean   admits a representation in terms of the Gauss hypergeometric function [13]: (see [13]).We will need the following.
Another result of interest (see [21]) reads as follows.
Theorem B. Let , V be positive numbers.Further, let  ≥ 1 and let  ≥ 1.Then the inequality holds true if and if Also, we will utilize the following result [22].
Theorem C. Let , V > 0 and assume that  ̸ = V.If V > 1, then

Inequalities
The goal of this section is to establish an inequality for the -version of the Schwab-Borchardt mean   and other inequalities as well.Applications of those results to generalized trigonometric and generalized hyperbolic functions are presented in the next section.We begin proving an extension of inequality (14).
Theorem 1.Let ,  > 0 ( ̸ = ) and let  > 1.Then where Proof.We need to prove the first inequality in (19).To this aim we will demonstrate first that This can be proven using the following upper bound for Gauss' hypergeometric function: which holds true if  > 0,  >  > 0, and || < 1 (see [23, (3.4), (2.15)]).Application to the Gauss hypergeometric function on the right side of (12) yields International Journal of Analysis 3 This in conjunction with (12) gives the desired inequality (21).
For the proof of the first inequality in (19) we apply (21) to the middle term of ( 19) to obtain where in the last step we have used (20).The proof is complete.
Our next result reads as follows.
Theorem 2. Let  and  be positive and unequal numbers.Also, let the number  be such that where 0 <  < 1.Then the following inequalities hold true.
Proof.We will prove now inequality (26).It follows from (25) followed by application of the inequality of arithmetic and geometric means, with weights  and 1 − , that ) Inequality ( 27) can be established in a similar manner.We use (26) again followed by a little algebra to obtain This yields where in the last step we have applied the Schwarz-Bunyakovsky inequality.

Applications to Generalized Trigonometric and Hyperbolic Functions
In this section we present several inequalities for the generalized trigonometric and hyperbolic 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.
The second Huygens and the second Wilker inequalities for the trigonometric functions also appear in mathematical literature.They read, respectively, as follows: (0 < || < /2).For the proofs of the last two results the interested reader is referred to [22,29], respectively.It is worth mentioning that there are known counterparts of inequalities ( 31)-(34) for the hyperbolic functions.They have the same structure as ( 31)-(34) with following modifications: sin → sinh and tan → tanh.The domains of their validity consist of all nonzero numbers.For more details and additional references see, for example, [22].