A New Method to Prove and Find Analytic Inequalities
Xiao-Ming Zhang,1Bo-Yan Xi,2 and Yu-Ming Chu1
1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2Department of Mathematics, Inner Mongolia University for the Nationalities, Tongliao 028000, China
Academic Editor: John Rassias
Received19 Oct 2009
Revised26 Jan 2010
Accepted02 Feb 2010
Published07 Apr 2010
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 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
Let (see (1.5)). (1) If , then
() If , then
From Theorem 1.2 we get
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
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
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
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.
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
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
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).