Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means

Abstract

For , the power mean of order of two positive numbers and is defined by , for , and , for . In this paper, we answer the question: what are the greatest value and the least value such that the double inequality holds for all and with ? Here , , and denote the classical arithmetic, geometric, and harmonic means, respectively.

1. Introduction

For , the power mean of order of two positive numbers and is defined by

(1.1)

Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities for can be found in literatures [1–12]. It is well known that is continuous and increasing with respect to for fixed and . Let , and be the classical arithmetic, geometric, and harmonic means of two positive numbers and , respectively. Then

(1.2)

In [13], Alzer and Janous established the following sharp double inequality (see also [14, page 350]):

(1.3)

for all .

In [15], Mao proved

(1.4)

for all , and is the best possible lower power mean bound for the sum .

The following sharp bounds for and in terms of power mean are proved in [16]:

(1.5)

for all .

The purpose of this paper is to answer the question: what are the greatest value and the least value such that the double inequality

(1.6)

holds for all and with ?

2. Main Result

In order to establish our main results we need the following lemma.

Lemma 2.1.

If , and , then

() for and ;

() for and .

Proof.

Simple computations lead to

(2.1)

(2.2)

where :

(2.3)

(2.4)

(2.5)

(2.6)

()If and , then (2.6) implies

(2.7)

Therefore, Lemma 2.1() follows from (2.1)–(2.3) and (2.5) together with (2.7).

()If and , then (2.6) yields

(2.8)

Therefore, Lemma 2.1(2) follows from (2.1)–(2.3) and (2.5) together with (2.8).

Theorem 2.2.

For all and with , one has

() for ;

(2) for , and for , each equality occurs if and only if , and and are the best possible power mean bounds for the product .

Proof.

• (1) If , then simple computations lead to

  

  (2.9)

() If and , then we clearly see that

(2.10)

If and , without loss of generality, we assume that . Let and , then , and simple computations lead to

(2.11)

(2.12)

Therefore, for follows from (2.11) and Lemma 2.1(1) together with (2.12), and for follows from (2.11) and Lemma 2.1(2) together with (2.12).

Next, we prove that and are the best possible power mean bounds for the product .

Firstly, we prove that is the best possible upper power mean bound for the product if .

For any and , one has

(2.13)

Let , then the Taylor expansion leads to

(2.14)

Equations (2.13) and (2.14) imply that if , then for any there exists , such that for .

Secondly, we prove that is the best possible lower power mean bound for the product if .

For any and , one has

(2.15)

From (2.15) and , we clearly see that

(2.16)

Equation (2.16) implies that if , then for any there exists , such that for .

Thirdly, we prove that is the best possible lower power mean bound for the product if .

For any and , one has

(2.17)

where −.

Let , then the Taylor expansion leads to

(2.18)

Equations (2.17) and (2.18) imply that if , then for any there exists , such that for .

Finally, we prove that is the best possible upper power mean bound for the product if .

For any and , one has

(2.19)

From (2.19) and we clearly see that

(2.20)

Equation (2.20) implies that if , then for any there exists , such that for .