Two Sharp Inequalities for Power Mean, Geometric Mean, and Harmonic Mean

Abstract

For , the power mean of order of two positive numbers and is defined by . In this paper, we establish two sharp inequalities as follows: and for all . Here and denote the geometric mean and harmonic mean of and respectively.

1. Introduction

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

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Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities for can be found in literature  [1–12]. It is well known that is continuous and increasing with respect to for fixed and . If we denote by and the arithmetic mean, geometric mean and harmonic mean of and , respectively, then

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

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for all

In [15], Mao proved

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

The purpose of this paper is to answer the questions: what are the greatest values and , and the least values and , such that and for all ?

2. Main Results

Theorem 2.1.

for all , equality holds if and only if , and is the best possible lower power mean bound for the sum .

Proof.

If , then we clearly see that .

If and , then simple computation leads to

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Next, we prove that is the best possible lower power mean bound for the sum .

For any and , one has

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where .

Let , then the Taylor expansion leads to

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Equations (2.2) and (2.3) imply that for any there exists , such that for .

Remark 2.2.

For any , one has

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Therefore, is the best possible upper power mean bound for the sum .

Theorem 2.3.

for all , equality holds if and only if , and is the best possible lower power mean bound for the sum .

Proof.

If , then we clearly see that

If and , then elementary calculation yields

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Next, we prove that is the best possible lower power mean bound for the sum .

For any and , one has

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where .

Let , then the Taylor expansion leads to

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Equations (2.6) and (2.7) imply that for any there exists , such that

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for .

Remark 2.4.

For any , one has

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Therefore, is the best possible upper power mean bound for the sum .