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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].
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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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Zhang, X., Chu, Y. A New Method to Study Analytic Inequalities. J Inequal Appl 2010, 698012 (2010). https://doi.org/10.1155/2010/698012
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Keywords
- Partial Derivative
- Convex Function
- Study Analytic
- Constant Factor
- Continuous Partial Derivative
