/ / Article

Research Article | Open Access

Volume 2010 |Article ID 128934 | 19 pages | https://doi.org/10.1155/2010/128934

# A New Method to Prove and Find Analytic Inequalities

Revised26 Jan 2010
Accepted02 Feb 2010
Published07 Apr 2010

#### 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 , 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  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 .

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 35 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  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 : Theorems 5.4 and 5.5 are the improvements of inequalities (5.24).

Theorem 5.4. If and , then

Proof. Let Then