Abstract
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. Let , be positive numbers. Further, let and let . Then the inequality holds true if and if
Also, we will utilize the following result [22].
Theorem C. Let and assume that . If , then
3. 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 and let . 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 , , and (see [23, (3.4), (2.15)]). Application to the Gauss hypergeometric function on the right side of (12) yields 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 . 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 , 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.
4. 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, 7–9, 13] and to the references therein.
In order to facilitate presentation we recall first some known results for classical trigonometric functions. In particular, the following results () have attracted attention of several researchers. Inequalities (31) and (32) have been obtained, respectively, by Huygens [24] and Wilker [25]. Several proofs of these results can be found in mathematical literature (see, e.g., [21, 22, 26–32] and the references therein). In [22] the authors called inequalities (31) and (32) the first Huygens and the first Wilker inequalities, respectively, for the trigonometric functions.
The second Huygens and the second Wilker inequalities for the trigonometric functions also appear in mathematical literature. They read, respectively, as follows: (). 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: and . The domains of their validity consist of all nonzero numbers. For more details and additional references see, for example, [22].
We are in a position to prove the following.
Theorem 3. Let . Then for all .
Proof. Let , where , and let . By making use of (11) and the formula (see [7]) we obtain Also, a use of (11) followed by application of (8) yields To obtain the desired result it suffices to apply inequality (19).
A particular case of (35), when , appears in [33].
It is worth mentioning that the counterpart of (35) for the generalized hyperbolic functions can also be established in a similar manner. An identity needed in the proof can be found in [7]. 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.
Theorem 4. Let and let . Then
Proof. We will employ Theorem 2 with , , and . This yields and . Inequality (41) follows from (26). Similarly, inequality (42) is an immediate consequence of (27) because .
Inequality (41) has been established by different means in [7, Theorem 3.16].
Our next result reads as follows.
Theorem 5. For the following inequalities hold true for all .
Proof. Inequality (43) is established in [7, Theorem 3.22]. It is included here for the sake of completeness. We will prove now inequality (44). To this aim we let It follows from [7, (3.7)] and the proof of Lemma 3.32 in [7] that holds for all . To obtain the desired result we apply now Theorem B, with and as defined above and and . It is easy to see that inequality (17) is the same as the second Huygens inequality (43). The assertion now follows.
The counterparts of Theorems 4 and 5 for the generalized hyperbolic functions can be established in a similar fashion. We leave it to the interested reader.
The last result of this section gives an inequality which connects the first and the second inequalities of Wilker for the generalized trigonometric functions.
Theorem 6. Let . If , then
Proof. For the sake of notation let Then the inequality connecting the first and the third members of (35) can be written as Exponentiation with the exponent of allows us to write the inequality connecting the first and the last members as . To complete the proof it suffices to utilize Theorem C with replaced by .
Conflict of Interests
The author declares that he had no conflict of interests.