A New Method to Study Analytic Inequalities

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.

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

(1.1)

If for all , then

(1.2)

for all .

Proof.

Without loss of generality, since we assume that and .

For , we clearly see that , then

(1.3)

From the continuity of the partial derivatives of and

(1.4)

we know that there exists such that and

(1.5)

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

(1.6)

Next, for , then and

(1.7)

Hence, we get

(1.8)

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

(1.9)

If for all , then

(1.10)

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

(1.11)

If for all and , then

(1.12)

for all with .

Corollary 1.4.

Suppose with , then

(1.13)

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

(1.14)

where . Equality holds if and only if .

Corollary 1.5.

Suppose with , has continuous partial derivatives and

(1.15)

If for all and , then

(1.16)

where . Equality holds if and only if .

Corollary 1.6.

Suppose with , then

(1.17)

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

(1.18)

where . Equality holds if and only if .

In this section, we denote , , and

(2.1)

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

(2.2)

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

(2.3)

Equality holds if and only if .

Proposition 2.2 (Holder Inequality).

Suppose that , . If , then

(2.4)

Proof.

Let and

(2.5)

If , then

(2.6)

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

(2.7)

Therefore, Proposition 2.2 follows from and .

Proposition 2.3 (Minkowski Inequality).

Suppose that ,. If , then

(2.8)

Proof.

Let and

(2.9)

If , then

(2.10)

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

(2.11)

Therefore, Proposition 2.3 follows from and .

If with , then the well-known Hardy's inequality (see [1,Theorem ]) is

(3.1)

In this section, we establish the following result involving Hardy's inequality.

Theorem 3.1.

Let , , and . If

(3.2)

then

(3.3)

Proof.

Let , then inequality (3.3) is equivalent to

(3.4)

and . Let

(3.5)

If , then

(3.6)

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

(3.7)

Then Theorem 1.1 leads to

(3.8)

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

Corollary 3.2.

Let , , and . If

(3.9)

then

(3.10)

Proof.

From inequality (3.3), we clearly see that

(3.11)

Remark 3.3.

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

If with , then the well-known Carleman's inequality is

(4.1)

with the best possible constant factor (see [2]).

Recently, Yang and Debnath [3] gave a strengthened version of (4.1) as follows:

(4.2)

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

(4.3)

(4.4)

Proof.

Let , then inequality is equivalent to inequality

(4.5)

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

If , then it is not difficult to verify that and

(4.6)

If , then ; this implies that

(4.7)

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

(4.8)

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

(4.9)

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

(4.10)

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

(4.11)

Theorem 4.2.

Let , , and . If , then

(4.12)

Proof.

Let , , and ,

(4.13)

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

(4.14)

where .

If , then

(4.15)

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

(4.16)

Therefore, inequality (4.14) is proved.

Corollary 4.3.

Let , , and . If , then

(4.17)

Proof.

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

(4.18)

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

Corollary 4.4.

If with , then

(4.19)

Remark 4.5.

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

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.

Affiliations

1. Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, China

   • Xiao-Ming Zhang
   •  & Yu-Ming Chu

Corresponding author

Correspondence to Yu-Ming Chu.

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