#### Abstract

We present a new method to study analytic inequalities. As for its applications, we prove the well-known Hölder inequality and establish several new analytic inequalities.

#### 1. Monotonicity Theorem

Throughout the paper denotes the set of real numbers and denotes the set of strictly positive real numbers. Let , and ; the arithmetic mean and the power mean of order with respect to the positive real numbers are defined by , for , and , respectively.

In [1], Pachpatte gave many basic methods and tools for researchers working in inequalities. In this section, we present a monotonicity theorem which can be used as powerful tool to prove and find analytic inequalities.

Lemma 1.1. *Suppose that , . If has continuous partial derivatives, then holds in if and only if holds for all and with .*

*Proof. *We only prove the case of .*Necessity*. For all and with , by the assumption we have . Then from the Langrange's mean value theorem we know that there exists such that
Letting , we get
According to the continuity of partial derivatives, we know that
holds also.*Sufficiency*. For all and with , from the assumption and the Langrange's mean value theorem we know that there exists such that

Therefore the proof of Lemma 1.1 is completed.

Theorem 1.2. *Suppose that is a symmetric convex set with nonempty interior, has continuous partial derivatives, and
**. If for all with ,
**
holds in , then
**
for all , with equality if only if .*

*Proof. *If , then Theorem 1.2 follows from Lemma 1.1 and . We assume that in the next discussion. Without loss of generality, we only prove the case of with .

If , then inequality (1.7) is clearly true. If , then without loss of generality we assume that .

(1) If and , then . From Lemma 1.1 and the conditions in Theorem 1.2 we know that there exist and such that , or , and
For the sake of convenience, we denote . Consequently,
If , then Theorem 1.2 holds. Otherwise, for the case of , and
From the continuity of partial derivatives we know that there exists such that
where and . Denote , , . By Lemma 1.1, we get
and . For the case of , after a similar argument, we get inequality (1.12) with .

Repeating the above steps, we get such that is a constant and are monotone increasing (decreasing) sequences if , and
If there exists such that , then the proof of Theorem 1.2 is completed. Otherwise, we denote ; without loss of generality, we assume that
where is a subsequence of . Then from the continuity of function , we get
If , then we repeat the above arguments and get a contradiction to the definition of . Hence . From we get ; the proof of Theorem 1.2 is completed.

() The proof for the case of or is implied in the proof of (1).

In particular, according to Theorem 1.2 the following corollary holds.

Corollary 1.3. *Suppose that is a symmetric convex set with nonempty interior, is a symmetric function with continuous partial derivatives, and
**
If holds in , then
**
for all , and equality holds if and only if .*

#### 2. Comparing with Schur's Condition

The Schur convexity was introduced by I. Schur [2] in 1923; the following Definitions 2.1 and 2.2 can be found in [2, 3].

*Definition 2.1. *For , without loss of generality one assumes that and . Then is said to be majorized by (in symbols if for and .

*Definition 2.2. *Suppose that A real-valued function is said to be Schur convex (Schur concave) if implies that .

Recall that the following so-called Schur's condition is very useful for detering whether or not a given function is Schur convex or concave.

Theorem 2.3 (see [2, page 57]). *Suppose that is a symmetric convex set with nonempty interior . If is continuous on and differentiable in , then is Schur convex (Schur concave) on if and only if it is symmetric and
**
holds for any .*

It is well known that a convex function is not necessarily a Schur convex function, and a Schur convex function need notbe convex in the ordinary sense either. However, under the assumption of ordinary convexity, is Schur convex if and only if it is symmetric [4].

Although the Schur convexity is an important tool in researching analytic inequalities, but the restriction of symmetry cannot be used in dealing with nonsymmetric functions. Obviously, Theorem 1.2 is the generalization and development of Theorem 2.3; the following results in Sections 3–5 show that a large number of inequalities can be proved, improved, and found by Theorem 1.2.

#### 3. A Proof for the Hölder Inequality

Using Theorem 1.2 and Corollary 1.3, we can prove some well-known inequalities, for example, power mean inequality, Hölder inequality, and Minkowski inequality. In this section, we only prove the Hölder inequality.

Proposition 3.1 (Hölder inequality). *Suppose that
**
If and , then
*

*Proof. *Let and
Then
Let (see (1.5)).

(1) If , then

() If , then
From Theorem 1.2 we get
that is,
Therefore, the Hölder inequality follows from (3.8) with and .

#### 4. Improvement of the Sierpiński Inequality

In the section, we give some improvements of the well-known Sierpiński inequality:

Theorem 4.1. *Suppose that , , . If for and for , then
*

*Proof. *Let , . Then
*Case 1. *. Let
Then
Therefore, is monotone increasing in . From
we know that , . Then leads to and
We assume that (see (1.16)). Let . Then inequality (4.8) becomes
Combining inequalities (4.3) and (4.9) yields that . Using Corollary 1.3 we have
Letting , we get
*Case 2. *. Let . Then from and , one has
Hence inequality (4.8) holds. The rest is similar to above, so we omit it.

The proof of Theorem 4.2 is similar to the proof of Theorem 4.1, and so we omit it.

Theorem 4.2. *Suppose that , , . If for and for , then
*

Theorem 4.3. *Suppose that , . If , then and
*

*Proof. *Let and
Then
Therefore, we get
We assume that (see (1.16)). Then we have
Letting , from , we get

According to Bernoulli's inequality with and , one has
For and , it is not difficult to verify that . Letting , we have
Using Corollary 1.3, we know that
Letting , we get

From (4.23), we get
Letting , we have
Inequality (4.14) is proved.

#### 5. Five New Inequalities

Let and . Then

References [5, 6] is the third symmetric mean of .

Theorem 5.1. *If , , then
**
with the best possible constant .*

*Proof. *Let and
Then
If (see (1.16)), then

Combining inequalities (5.5) and (5.6) yields that . Then from Corollary 1.3 we have
for all , which implies that
Therefore, inequality (5.2) is proved.

Taking and in inequality (5.2), we get

Letting , we get

So is the best possible constant.

For , , Alzer [7] established the following inequality:

Theorems 5.2 and 5.3 are the improvements of Alzer's inequality.

Theorem 5.2. *If , then
*

*Proof. *Firstly, let , and
Then
If , , then
Then from Corollary 1.3, we get
Let in above inequality. Then we know that inequality (5.12) holds. From continuity we know that inequality (5.12) holds also for .

Theorem 5.3. *If , then
*

*Proof. *Let
Then
If and , then from and we get

Let . Then
Thus is a monotone increasing function. This monotonicity and
lead to . Therefore and is a monotone increasing function. From and the monotonicity of we know that . By (5.20), we know that . According to Corollary 1.3 we get
Finally, let in the above inequality. Then we know that Theorem 5.3 holds.

If and , then the following inequalities can be found in [8–10]:
Theorems 5.4 and 5.5 are the improvements of inequalities (5.24).

Theorem 5.4. *If and , then
*

*Proof. *Let
Then
We assume that (see (1.16)). Then
According to Corollary 1.3, we get

Let
A similar argument as above leads to

The proof of Theorem 5.4 is completed.

Let

The proof of Theorem 5.5 is similar to the proof of Theorem 5.4, and so we omit it.

Theorem 5.5. *Let , . Then
*

*Remark 5.6. *More applications for Theorem 1.2 and Corollary 1.3 were shown in [11].

#### Acknowledgments

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. This research is partly supported by the N. S. Foundation of China under Grant no. 60850005, N. S. Foundation of Zhejiang Province under Grants nos. D7080080 and Y607128, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant no. T200924.