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# A New Method to Study Analytic Inequalities

*Journal of Inequalities and Applications***volume 2010**, Article number: 698012 (2010)

## Abstract

We present a new method to study analytic inequalities involving variables. Regarding its applications, we proved some well-known inequalities and improved Carleman's inequality.

## 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.

Suppose that with and , has continuous partial derivatives and

If for all , then

for all .

Proof.

Without loss of generality, since we assume that and .

For , we clearly see that , then

From the continuity of the partial derivatives of and

we know that there exists such that and

for any . Hence, since is strictly monotone increasing, then we have

Next, for , then and

Hence, we get

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.

Suppose that with and , has continuous partial derivatives and

If for all , then

for all .

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

Corollary 1.3.

Suppose that with , has continuous partial derivatives and

If for all and , then

for all with .

Corollary 1.4.

Suppose with , then

and is symmetric with continuous partial derivatives. If for all , then

where . Equality holds if and only if .

Corollary 1.5.

Suppose with , has continuous partial derivatives and

If for all and , then

where . Equality holds if and only if .

Corollary 1.6.

Suppose with , then

and is symmetric with continuous partial derivatives . If for all , then

where . Equality holds if and only if .

## 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.

It is well known that is symmetric with respect to and is continuous. Without loss of generality, we assume that . Then

If , then . It follows from Corollary 1.4 that we get

Equality holds if and only if .

Proposition 2.2 (Holder Inequality).

Suppose that , . If , then

Proof.

Let and

If , then

Similarly, if , then . From Theorem 1.1, we get

Therefore, Proposition 2.2 follows from and .

Proposition 2.3 (Minkowski Inequality).

Suppose that ,. If , then

Proof.

Let and

If , then

Similarly, If , then . It follows from Theorem 1.1 that we get

Therefore, Proposition 2.3 follows from and .

## 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.

Let , , and . If

then

Proof.

Let , then inequality (3.3) is equivalent to

and . Let

If , then

Making use of the well-known Hadamard's inequality of convex functions, we get

Then Theorem 1.1 leads to

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

Corollary 3.2.

Let , , and . If

then

Proof.

From inequality (3.3), we clearly see that

Remark 3.3.

If , then inequality (3.1) follows from inequality (3.10).

## 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.

If and , then

Proof.

Let , then inequality is equivalent to inequality

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

If , then it is not difficult to verify that and

If , then ; this implies that

From inequalities (4.6) and (4.7), we get

From the well-known Stirling Formula (), we get

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

Therefore, inequality (4.4) follows from inequalities (4.10) and

Theorem 4.2.

Let , , and . If , then

Proof.

Let , , and ,

Then inequality (4.12) is equivalent to the following inequality:

where .

If , then

From inequality (4.3) and together with Theorem 1.1, we clearly see that

Therefore, inequality (4.14) is proved.

Corollary 4.3.

Let , , and . If , then

Proof.

Let (), then inequality (4.4) implies that is a strictly increasing sequence. Then from inequality (4.12) we get

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

Corollary 4.4.

If with , then

Remark 4.5.

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

## References

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## 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.

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### Keywords

- Partial Derivative
- Convex Function
- Study Analytic
- Constant Factor
- Continuous Partial Derivative