On the -Version of the Schwab-Borchardt Mean

Abstract

This paper deals with a one-parameter generalization of the Schwab-Borchardt mean. The new mean is defined in terms of the inverse functions of the generalized trigonometric and generalized hyperbolic functions. The four new bivariate means are introduced as particular cases of the -version of the Schwab-Borchardt mean. For the particular value of the parameter , these means become either the classical logarithmic mean or the Seiffert means or the Neuman-Sándor mean. Wilker- and Huygens-type inequalities involving inverse functions of the generalized trigonometric and the generalized hyperbolic functions are also established.

1. Introduction

The Schwab-Borchardt mean of two numbers and , denoted by , is defined as (see [1, Thrm 8.4], [2, (2.3)]). It follows from (1) that is not symmetric in its arguments and is a homogeneous function of degree 1 in and . This mean has been studied extensively in [1–5].

The goal of this paper is to define and investigate a generalization of the Schwab-Borchardt mean SB. The new one has a form which is similar to (1) but depends on a parameter which is used in definitions of two families of higher transcendental functions called the generalized trigonometric and the generalized hyperbolic functions. Definitions of these functions are given in Section 2. Also, we will use a notion of the -hypergeometric functions of two variables. Their definition and some basic properties are given in Section 3. Throughout the sequel, the of SB will be denoted by . The latter is introduced in Section 4. Therein, some basic properties of the new mean are discussed. In Section 5, we define four new bivariate means which can be considered as the generalized logarithmic mean, the generalized Seiffert means, and the generalized Neuman-Sándor mean. In the last section of this paper, we shall establish inequalities involving new means as well as Wilker- and Huygens-type inequalities involving inverse functions of the generalized trigonometric and the generalized hyperbolic functions.

2. Definitions of Generalized Trigonometric and Hyperbolic Functions

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 .

In what follows, we will always assume that the number is strictly greater than 1. We will adopt notation and definitions used in [6]. Let Further, let Also, let and let . The generalized trigonometric and hyperbolic functions needed in this paper are the following homeomorphisms: The inverse functions of and can be represented as follows [7]:

Inverse functions of the remaining four functions can be expressed in terms of and . We have

The generalized trigonometric functions have been introduced by Lindqvist in [8]. It is obvious that the functions under discussion 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 generalized trigonometric functions, generalized hyperbolic functions, and inequalities involving these functions, the interested reader is referred to [6–15].

3. The -Hypergeometric Functions of Two Variables

In this section, we give the definition of the bivariate -hypergeometric functions which are used in the sequel. Some results for these functions are also included here.

In what follows, the symbols and will stand for the nonnegative semiaxis and the set of positive numbers, respectively. Let . By , where we will denote the Dirichlet measure on the interval . It is well known that is the probability measure on its domain.

Also, let . In [16, 17], the -hypergeometric function () is defined as follows: where and are the dot product of and . Many of the important special functions, including Gauss’ hypergeometric function and some elliptic integrals, admit the integral representation (13). For more details, the interested reader is referred to Carlson’s monograph [17].

A nice feature of the -hypergeometric function is its permutation symmetry in both parameters and variables; that is, Another remarkable property of is its homogeneity of degree in its variables: .

For the later use, let us also record Carlson’s inequality [18, Theorem 3]: (, ).

We will also need the following result which appears in [5, Proposition 2.1]. Let , , and let . Then the following inequality holds true for all .

4. Definition and Basic Properties of the -Version of SB

Let the numbers and have the meaning as in Section 1. For the sake of presentation, we recall first a formula for the mean in terms of the -hypergeometric function: (see [17, 19]).

We define the -version of the mean as follows:

The rightmost member of (19) is a special case of what is called in mathematical literature the -hypergeometric mean (see [2, 17, 18]). Using elementary properties in the -hypergeometric functions, we see that is the mean value of and . Moreover, this mean is nonsymmetric and homogeneous of degree 1 in its variables. The well-known results on the -hypergeometric means lead to the conclusion that is a strongly increasing function of the parameter .

For the brevity of notation, let us introduce a particular -hypergeometric function: Clearly function is nonsymmetric and homogeneous of degree in its variables. Comparison with (19) yields

We shall demonstrate now that can also be expressed in terms of and :

For the proof of the first part of (22), let us record a formula which shows that the Gauss hypergeometric function can be expressed in terms of the bivariate -hypergeometric function:

(see, e.g., [17, ((5.9)–(12))]). Application of the last formula to (7) yields where . This, in conjunction with (9), gives Letting above and utilizing homogeneity of the function , we obtain where in the last step we have utilized formula (19). This yields the first part of (22). The second part can be established in an analogous manner. A key formula needed here reads as follows: . We omit further details.

Function admits an integral representation: This follows from the known result [20, (19.16.9)] where stands for the beta function and . Letting and , we obtain . Formula (28) now follows because .

5. Four New Bivariate Means Derived from SB_(p)

The goal of this section is to define and investigate four new bivariate means. They are defined in terms of the and the bivariate power mean . Recall that where . The power mean of order is usually denoted by and is called the geometric mean. It is well known that the power mean is a strictly increasing function of .

We are in a position to define four new means of positive numbers and . In what follows, these means will be denoted by and , where . They are defined as follows:

In the case when , these means become the classical logarithmic mean , two Seiffert means and (see [21, 22]), and the Neuman-Sándor mean introduced in [4].

For the later use, we introduce quantity , where

Clearly, .

The main result of this section reads as follows.

Theorem 1 (let .). Then the following formulas are valid.

Proof. We begin with the proof of (36). Making use of (31) and (22), we obtain Elementary computations yield Multiplying and dividing by we obtain, using (35), We shall write the denominator of (40) using (10) and (43) as follows: This, in conjunction with (40) and (43), gives the desired result (36).
We shall provide now a sketch of the proof of formula (39). It follows from (22) and (34) that Elementary computations yield
Application of (10) with gives . This in conjunction with (45) and (46) yields the asserted result (39). The remaining two formulas for the -versions and of the Seiffert means can be established in an analogous manner using (32) or (33), (22), and (9). We leave it to the interested reader. The proof is complete.

6. Inequalities Involving the SB_(p) Means

This section deals with inequalities involving the means. In particular, inequalities for the four means introduced in Section 5 are established. Also, we shall prove Wilker-type and Huygens-type inequalities involving inverse functions of the generalized trigonometric functions and the generalized hyperbolic functions.

Our first result reads as follows.

Theorem 2. Let the positive numbers and be such that . Then

Proof. We shall prove the assertion using integral formula (28) and formula (21). Let and let . Then and or what is the same that because . Integration yields or what is the same that the inequality Raising both sides to the power of −1 and next applying formula (21), we obtain Letting and next utilizing homogeneity of , we obtain the desired result.

The four new means defined in Section 5 and the power means are comparable. We have the following.

Corollary 3. Let , , , , and be the mean values of two positive and unequal numbers. Then the following chain of inequalities is valid.

Proof. The first, third, fourth, and the sixth inequalities in (53) follow from their definitions (see (31), (32), (34), and (33)). The second and the fifth inequalities can be obtained using Theorem 2 applied to two pairs of defining equations (31)-(32) and (34)-(33).

Our next result reads as follows.

Theorem 4. Let and . Then the inequality holds true for positive and unequal numbers and .

Proof. We shall obtain the desired result utilizing the following transformation for the -hypergeometric functions [17, ((5.5)–(19))]: where . Letting and and making substitutions and , we obtain Applying the permutation symmetry (see (14)) to the -hypergeometric function on the left-hand side and next raising both sides of the resulting formula to the power of −1, we obtain Since , Carlson’s inequality (16) yields Combining this with (57), we obtain Utilizing (20) and (21), we can write the last inequality in the form The desired inequality (54) now follows.

Corollary 5. Let the numbers and be the same as in Theorem 4. Then the following inequalities hold true.

Proof. Inequality (61) can be obtained using Theorem 4 with and . Next, we utilize formulas (31) and (32) to obtain the desired result. In a similar fashion, one can prove inequality (62) using Theorem 4 with and . Making use of formulas (31) and (32) yields the assertion. The remaining inequalities (63) can be proven in an analogous manner. We omit further details.

Another inequality for the mean is contained in the following.

Theorem 6. Let . Then

Proof. First, we make the substitutions in (17) next we use (20), and finally we raise both sides of the resulting inequality to the power of −2. This gives Application of (21) gives the desired inequality (64).

Corollary 7. Assume that the positive numbers and are not equal. Then the following inequalities hold true.

Proof. For the proof of (67), we let in (64) , and . Then, the left-hand side of (67) follows from (32) and (34). To obtain the right-hand side of the inequality in question, let us notice that and obviously . Inequality (68) can be established in a similar manner. We use (64) with , and followed by application of (31) and (33).

We will close this section proving Wilker-type and Huygens-type inequalities which involve inverse functions of the generalized trigonometric and hyperbolic functions. To this aim, we shall employ the following result [23].

Theorem A. Let be positive numbers. Assume that and satisfy the separation condition: Then the inequality holds true if either or for some with . If and satisfy the separation condition (69) together with then inequality (70) is also valid if

As in the previous sections, the letters and will stand for positive and unequal numbers. Also, and denote the power means of and .

For the sake of notation, we define

We are in a position to prove the following.

Theorem 8. Let be the same as in (35) and let . Assume that . Also, let or If then inequality (70) is satisfied if is defined either in (76) or in (77). Inequality (70) is also satisfied if either or provided that Here, .

Proof. For the proof of validity of (70) with as defined in (76), we let It follows from (53) that the separation condition (69) is satisfied. To complete the proof of (76), we utilize a well-known fact about bivariate means. Let be a mean which is homogeneous of degree 1 in its variables. Then In particular, we have This together with (36) and (37) gives the explicit formula (76) for . We will show now that the first inequality in (71) is satisfied if and . To this aim, we utilize (82) and write inequality (61) as follows: where and . This yields and . To obtain condition (78) of validity of (70), we substitute into the last inequality in (71). This completes the first part of the proof. Assume now that is the same as is defined in (79). We will prove that (70) holds true provided that condition (81) is satisfied. First, we define Again, we appeal to (53) to claim that and satisfy the separation condition (69). Making use of (36) and (37), we obtain an explicit formula (79) for . We will show that the first inequality in (71) is satisfied if and . To this aim, we utilize (86) and write inequality (62) as follows: where and . To prove that (70) holds true if (81) is satisfied, we substitute into the last inequality in (71). The assertion now follows. The remaining two cases when is defined in (77) or in (80) can be established in the analogous manner. In these cases, we have either or We leave it to the reader to complete the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.