# A New Method to Study Analytic Inequalities

- Xiao-Ming Zhang
^{1}and - Yu-Ming Chu
^{1}Email author

**2010**:698012

https://doi.org/10.1155/2010/698012

© X.-M. Zhang and Y.-M. Chu. 2010

**Received: **16 October 2009

**Accepted: **24 December 2009

**Published: **5 January 2010

## Abstract

## 1. Monotonicity Theorems

Throughout this paper, we denote the set of real numbers and the set of strictly positive real numbers, , .

In this section, we present the main results of this paper.

Theorem 1.1.

Proof.

Without loss of generality, since we assume that and .

For , we clearly see that , then

If , then Theorem 1.1 is true. Otherwise, we repeat the above process and we clearly see that the first and second variables in are decreasing and no less than . Let be their limit values, respectively, then and . If , then Theorem 1.1 is also true; otherwise, we repeat the above process again and denote and the greatest lower bounds for the first and the second variables , respectively. We clearly see that ; therefore, and Theorem 1.1 is true.

Similarly, we have the following theorem.

Theorem 1.2.

It follows from Theorems 1.1 and 1.2 that we get the following Corollaries 1.3–1.6.

Corollary 1.3.

Corollary 1.4.

where . Equality holds if and only if .

Corollary 1.5.

where . Equality holds if and only if .

Corollary 1.6.

## 2. Unifying Proof of Some Well-Known Inequality

In this section, we denote , , and

Proposition 2.1 (Power Mean Inequality).

If the power mean of order is defined by for and , then for ; equality holds if and only if .

Proof.

Equality holds if and only if .

Proposition 2.2 (Holder Inequality).

Proof.

Therefore, Proposition 2.2 follows from and .

Proposition 2.3 (Minkowski Inequality).

Proof.

## 3. A Brief Proof for Hardy's Inequality

If with , then the well-known Hardy's inequality (see [1,Theorem ]) is

In this section, we establish the following result involving Hardy's inequality.

Theorem 3.1.

Proof.

and we clearly see that inequalities (3.4) and (3.3) are true.

Corollary 3.2.

Proof.

Remark 3.3.

## 4. A Refinement of Carleman's Inequality

If with , then the well-known Carleman's inequality is

with the best possible constant factor (see [2]).

Recently, Yang and Debnath [3] gave a strengthened version of (4.1) as follows:

Some other strengthened versions of (4.1) were given in [4–9]. In this section, we give a refinement for Carleman's inequality (see Corollary 4.4).

Lemma 4.1.

Proof.

If , then simple computation leads to inequality (4.5).

If , then it is not difficult to verify that and

Therefore, inequality (4.5) follows from inequalities (4.8) and (4.9).

From the monotonicity of sequence and , we get ; therefore, inequality (4.3) is proved.

Meanwhile, we have

Theorem 4.2.

Proof.

Therefore, inequality (4.14) is proved.

Corollary 4.3.

Proof.

Let ; thus, we know that Corollary 4.4 is true.

Corollary 4.4.

Remark 4.5.

Many other applications for Theorem 1.1 appeared in [10].

## Declarations

### Acknowledgments

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. This work was partly supported by the National Nature Science Foundation of China under Grant no. 60850005, the Nature Science Foundation of Zhejiang Province under Grant no. Y607128, the Nature Science Foundation of China Central Radio & TV University under Grant no. GEQ1633, and the Nature Science Foundation of Zhejiang Broadcast & TV University under Grant no. XKT-07G19.

## Authors’ Affiliations

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## Copyright

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