On the Inequalities for the Generalized Trigonometric Functions

This paper deals with Huygens-type and Wilker-type inequalities for the generalized trigonometric functions of P. Lindqvist. A major mathematical tool used in this work is a generalized version of the Schwab-Borchardt mean introduced recently by the author of this work. 1. 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 . They become classical trigonometric and hyperbolic functions when . 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 [1–11]. 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. 2. Definitions and Preliminaries For the reader’s convenience we recall first definition of the celebrated Gauss hypergeometric function : where is the shifted factorial or Appell symbol, with if , and (see, e.g., [12]). In what follows, let the parameter be strictly greater than 1. In some cases this assumption will be relaxed to . We will adopt notation and definitions used in [5]. Let Further, let Also, let and let . The generalized trigonometric and hyperbolic functions needed in this paper are the following homeomorphisms: The inverse functions and are represented as follows [7]: Inverse functions of the remaining four functions can be expressed in terms of and . We have For the later use we recall now definition of a certain bivariate mean introduced recently in [13] and call the -version of the Schwab-Borchardt mean. When , the latter mean becomes a classical Schwab-Borchardt mean which has been studied extensively in [14–20]. It is clear that is a nonsymmetric and homogeneous function of degree 1 of its variables. 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. Theorem A. If , then Let and let . Then the inequality holds true for all positive and unequal numbers and (see [13]). Another result of interest (see [21]) reads as follows. Theorem B.