On the -Version of the Schwab-Borchardt Mean II

Abstract

This paper deals with the -version of the Schwab-Borchardt mean. Lower and upper bounds for this mean, expressed in terms of the weighted geometric and arithmetic means of its variables, are obtained. Applications to four bivariate means, introduced earlier by the author of this paper, are included.

1. Introduction

In the past few years, there is a renewed interest in investigations of bivariate means. In this paper, we deal with two-sided 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 [2–4]. 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.

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 the Appell symbol, with if and (see, e.g., [5]).

In what follows, we will assume that the parameter is strictly greater than 1. In some cases, this assumption will be relaxed to . We will adopt notation and definitions used in [6]. Let Further, let Also, 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 two functions can be expressed in terms of and . 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., [9–12]) as well.

For the later use, we recall now definition of a certain bivariate mean introduced recently in [1] And we call the -Schwab-Borchardt mean. When , 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 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 [1]:

We close this section with definitions of some bivariate means used in the sequel. To this end, we will assume that . The power mean    of order of and is defined in a usual way as The power mean 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 (), they have been introduced in [1], and they are defined as follows:

In the case when , 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]:

3. Main Results

First, we will deduce lower and upper bounds for the mean . They have form of the weighted geometric and arithmetic means of and .

Throughout the sequel, we will always assume that the parameter satisfies 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 . Otherwise, if , then 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 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 with optimal value . The upper bound for 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] () and (23), we obtain the desired result. We will establish now bounds (18) which are valid provided that and . To this aim, we use a second part of (8) to obtain Letting and taking into account that (see (7)), we obtain The lower bound for is Taking into account that   , inequality (31) can be written as It has been demonstrated in [7, Theorem 3.8] that the last inequality is satisfied for all 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 and if . Formula (19) now follows and the proof is complete.

Bounds (17) and (18), when , have been obtained in [3].

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 or . The upper bound is the weighted arithmetic means of the same pairs of elementary bivariate means.

We have the following.

Proposition 2. Let the numbers and be the same as in (16) and (19). Then, where the right inequalities in (35) and (36) hold true provided that . Also,

Proof. In order to establish inequality (35), we use (12) and next apply (17) with and . The remaining inequalities (36)-(37) can be proven in an analogous manner. We omit further details.

Conflict of Interests

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