On the p-Version of the Schwab-Borchardt Mean II

In the past few years, there is a renewed interest in investigations of bivariate means. In this paper, we deal with twosided bounds for so-called p-Schwab-Borchardt mean and four special means generated by this one-parameter mean. All thesemeans have been introduced and investigated in our recent paper [1]. A special case of the p-Schwab-Borchardt mean is discussed in the author’s earlier papers [2–4]. This work, which is continuation of [1], is organized as follows. Definitions of generalized trigonometric and hyperbolic functions are included in Section 2. Main results of this paper are established in Section 3.


Introduction
In the past few years, there is a renewed interest in investigations of bivariate means.In this paper, we deal with twosided bounds for so-called -Schwab-Borchardt mean and four special means generated by this one-parameter mean.All these means have been introduced and investigated in our recent paper [1].A special case of the -Schwab-Borchardt mean is discussed in the author's earlier papers [2][3][4].This work, which is continuation of [1], is organized as follows.Definitions of generalized trigonometric and hyperbolic functions are included in Section 2. Main results of this paper are established in Section 3.
In what follows, we will assume that the parameter  is strictly greater than 1.In some cases, this assumption will be relaxed to 1 <  ≤ 2. We will adopt notation and definitions used in [6].Let Further, let Also, let  = (0, 1).The generalized trigonometric and hyperbolic functions needed in this paper are the following homeomorphisms: sin  : (0,   ) → , cos  : (0,   ) → , The inverse functions sin −1  and sinh −1  are represented as follows [7]: 2 International Journal of Mathematics and Mathematical Sciences Inverse functions of the remaining two functions can be expressed in terms of sin −1   and sinh −1  .We have Generalized trigonometric functions recently have attracted attention of several researchers.The interested reader is referred to a highly cited paper by Lindqvist [8] and other papers (see, e.g., [9][10][11][12]) as well.
For the later use, we recall now definition of a certain bivariate mean introduced recently in [ And we call SB  (, ) the -Schwab-Borchardt mean.When  = 2, the latter mean becomes a classical Schwab-Borchardt mean which has been studied extensively in mathematical literature (see, e.g., [2-4, 13, 14]).It is clear that SB  (, ) is a nonsymmetric and homogeneous function of degree 1 of its variables.
A remarkable result states that the mean SB  admits a representation in terms of the Gauss hypergeometric function [1]: We close this section with definitions of some bivariate means used in the sequel.To this end, we will assume that ,  > 0. The power mean   (, ) ( ∈ R) of order  of  and  is defined in a usual way as The power mean  0 is usually denoted by .It is well known that the power mean   is a strictly increasing function of .
We will deal with a quadruple of bivariate means of  and .They are denoted by   ,   ,   , and   ( > 1), they have been introduced in [1], and they are defined as follows: (, ) ≡   = SB  ( /2 ,   ) , In the case when  = 2, these means become the classical logarithmic mean , two Seiffert means  and  (see [15,16]), and the Neuman-Sándor mean  introduced in [3].
It is worth mentioning that the means defined above satisfy the chain of inequalities [1]:

Main Results
First, we will deduce lower and upper bounds for the mean SB  .They have form of the weighted geometric and arithmetic means of  and .
Throughout the sequel, we will always assume that the parameter  satisfies  > 1 unless otherwise stated.Also, let We will prove now the following.
Proposition 1.Let  and  be positive and unequal numbers.If  < , then the two-sided inequality holds true, where the second inequality in ( 17) is valid only if where Proof.Suppose that  < .Using the first part of (8), we get Letting and next applying the formula (see, e.g., [6] or ( 6)), we obtain The lower bound for SB  (, ) is the weighted geometric mean of  and ; that is, Applying (21) to the left side of ( 24) and (23) to the right side of the last inequality yields We appeal now to Theorem 3.6 in [7] to conclude that the last inequality is satisfied for all  ∈ (0,   ] with optimal value  =   .The upper bound for SB  (, ) is a weighted arithmetic mean of  and ; that is, Using ( 21) and ( 23), we can write the last inequality as follows: Utilizing inequality (3.23) in [7] sin (1 <  ≤ 2) and ( 23), we obtain the desired result.We will establish now bounds (18) which are valid provided that  >  and  > 1.To this aim, we use a second part of ( 8) to obtain Letting / = cosh   and taking into account that  √(cosh  )  − 1 = sinh   (see ( 7)), we obtain The lower bound for SB  (, ) is Taking into account that / = cosh   ( > 0), inequality (31) can be written as It has been demonstrated in [7,Theorem 3.8] that the last inequality is satisfied for all  ∈ (0,   ] with an optimal value if  =   .This completes the proof of the left inequality in (18).
To obtain the right-hand side inequality in (18), we employ the following inequalities: if  ≥ 2. Formula (19) now follows and the proof is complete.
We will apply now Proposition 1 to obtain two-sided bounds for the four bivariate means   ,   ,   , and   defined in Section 2. The lower bounds are weighted geometric means of either (,  /2 ) or ( /2 ,   ).The upper bound is the weighted arithmetic means of the same pairs of elementary bivariate means.
We have the following.