Journal of Mathematics
Volume 2016 (2016), Article ID 4346712, 11 pages
http://dx.doi.org/10.1155/2016/4346712

Research Article

Multiple-Term Refinements of Young Type Inequalities

Daeshik Choi

Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Box 1653, Edwardsville, IL 62026, USA

Received 2 July 2016; Accepted 10 October 2016

Academic Editor: Shaofang Hong

Copyright © 2016 Daeshik Choi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Recently, a multiple-term refinement of Young’s inequality has been proved. In this paper, we show its reverse refinement. Moreover, we will present multiple-term refinements of Young’s inequality involving Kantorovich constants. Finally, we will apply scalar inequalities to operators.

1. Introduction

The classical Young inequality states that if and , then

For , we define three functions , , and by

In [1] and the references there, the following improvements of Young inequality and its reverse are discussed:

where is the characteristic function defined by

Another form of Young type inequalities discussed in [1] is as follows:

Other types of improvements of the Young inequality is to use Kantorovich constants. Wu and Zhao [2] showed

where

Note that (7) improves (3), since for all . In [3], Liao and Wu improved (4) as follows:

where

The constants of the form are called Kantorovich constants.

Throughout the paper, we will use the following functions.

Definition 1. One defines the sequence of functions on as follows:

for .

Definition 2. For and , we define the functions by

The following multiple-term refinement of Young inequality has been recently proved in [1]. In the next section, we will present a different and simpler proof of it.

Theorem 3. Let and be any positive integer. Then

for .

The above is a simplified expression of the original one in [1] which is written in rather complicated notation, but they are essentially identical. Note that the first inequalities of (3) and (4) are obtained from (13) with and , respectively.

The object of this paper is to show

(1)a reverse of (13) which generalizes the second inequalities of (3) and (4),(2)multiple-term refinements of Young inequality involving Kantorovich constants which generalize (7) and (9),(3)operator inequalities related to Young inequality.

2. Multiple-Term Reverse of Young Inequality

From now on, we will fix and use the following functions:

for and an integer . As we will see (Lemma 4), for any integer with . Thus the interval of the characteristic function in or can include boundary points. For example, may be replaced by or .

We can express and as multipart functions as follows:

For any , we can formulate explicitly.

Lemma 4. Let and be integers. If , then

Proof. We prove it by induction on . The case is obvious. Assume that . If is odd, then and by induction. Since , one has

If is even, then and by induction. Since , we have

Using the same argument, we can show that if , then . We omit the detailed proof.

Lemma 5. For a positive integer , is the linear interpolation of at for .

Proof. Since is a line segment on each interval for , is a line segment on for . Thus it suffices to show

Note that since at , (19) holds for . We will prove (19) by induction on . Since , one has

Assume that (19) holds and . If is even, then

If is odd, then

Since is the line segment joining and on , we have

By Lemma 4, . Noting that , we can write by

Thus, from (22), we deduce that

Remark 6. Since is concave, by Lemma 5, which proves Theorem 3 in a much simpler way, where the original proof is done by mathematical induction on .

Lemma 7. Let be the reflection of about the point ; that is,

Then each of the following is true.

(1) for all .(2) is the linear interpolation of at at .

Proof. Let . Since

one derives that for , for , and . Thus we have for all . So part (1) is proved.
For the second part, it suffices to show that is the reflection of about the point ; that is,

Noting that and , one gets that

Theorem 8. For any integer and , one has

That is,

Proof. As mentioned in Remark 6, follows from the concavity of . Since and is the linear interpolation of which is convex on , .

Note that the second inequalities of (3) and (4) can be obtained from with and , respectively. Moreover, (6) follows from the following result with .

Corollary 9. For any integer and , one has

Proof. Replacing and by their squares in Theorem 8, we obtain that

Since for all , we derive from the above that

Hence (32) follows from the identity

3. Young Inequalities Involving Kantorovich Constants

In this section, we will discuss multiple-term improvements of Young inequality involving Kantorovich constants. For a nonnegative integer , we define by

Lemma 10. For , one has

Proof. Replacing by , the inequality is equivalent to

for . Taking the natural logarithm, it suffices to show that

A direct computation shows that

Thus for any .

Lemma 11. For a positive integer and , define by

for . Then one has

Proof. As mentioned in the previous section, since for , in the definition of may be replaced by . We prove (42) by induction on . We have

Suppose that (42) holds. Then

Replacing in the first summation and in the second summation, respectively, by , one has

Thus (42) holds for all positive integers .

The following shows a multiple-term refinement of Young inequality involving Kantorovich constants.

Theorem 12. For , , and , one has

Proof. Putting , (46) can be rewritten as

By Lemma 11, the above can be expressed by

Thus it suffices to show that if , then

for . Replacing by and letting , the above is equivalent to

Let . By Lemma 10, one gets that

Since

by Lemma 4, it follows that if , then and therefore (50) holds.

Note that (46) can be written as

which gives the first inequalities of (7) and (9) with and , respectively.

Now we consider a reverse inequality corresponding to Theorem 12. A given inequality of the form can be utilized to derive its reverse in many cases. For example, replacing by in

which is the first inequality of (3), we obtain

Since , the above implies the second inequality of (3). Similarly, replacing by in

which is the first inequality of (4), we get

Since , the above implies the second inequality of (4). In the same way, the first inequality in Theorem 8 can be used to derive

which is stronger than the second inequality in the theorem. Based on such an observation, we can show a reverse inequality corresponding to Theorem 12 as follows.

Theorem 13. For , , and , one has

Proof. Replacing by and by in (46), we have

where the last inequality results from

Note that the second inequalities of (7) and (9) follow from the above theorem with and , respectively.

4. Operator Inequalities

From now on, we use uppercase letters for invertible positive operators on a Hilbert space and lowercase letters for real numbers. The following notations will be used:

(i) () denotes that is a positive (invertible positive) operator.(ii) denotes that is a positive (invertible positive) operator.

For and , the -arithmetic and -geometric means of and are defined, respectively, by

In the case , we will omit the -value in them. For example, denotes .

The operator version of (1) is well known as follows:

for and positive invertible operators and (see [4, 5] for more matrix Young inequalities). To show operator inequalities corresponding to their scalar versions, we will use the operator monotonicity of continuous functions; that is, if is a real valued continuous function defined on the spectrum of a self-adjoint operator , then for every in the spectrum of implies that is a positive operator.

From now on, will denote the operator version of defined in Definition 2. That is,

for .

Theorem 14. Let and . Then

Proof. For any , we have

by Theorem 8. Thus, for any positive operator , we have

by the operator monotonicity of continuous functions, where is the identity operator. Note that since , we can express and by

Letting and then multiplying all terms by on both sides, (67) yields (65), where can be obtained as follows:

The following shows matrix inequalities corresponding to Corollary 9.

Theorem 15. Let be complex matrices. If and are positive semidefinite, then

where denotes the Frobenius norm and

Proof. Since and are positive semidefinite, there exist unitary matrices and such that and , where denotes the diagonal matrix whose th diagonal entry is . Letting , we can show the following directly:

Since is unitarily invariant, one obtains that

Thus, by (32), we have

Since

we have

Next we consider operator inequalities related to Kantorovich constants.

Theorem 16. Let and be a real number with . If either or , then

Proof. Letting in Theorem 12, we have

for any . Since is an increasing function on and for , for or . Thus, from (78), we have

for or . Replacing by which satisfies or , we get that

Multiplying the inequality by on both sides, we have

Since the zeroth term in the summation is , we finish the proof of the first inequality of this theorem.
We can prove the second one in the same way. Letting in Theorem 13, we have

for any . In particular, if or , then

Replacing by , one has

Finally, multiplying each term by on both sides, we get that

Since , the above shows the second inequality of this theorem.

In the following, we consider special values of .

Corollary 17. Let and . If for some , then

If for some , then

Proof. If , then . Thus from (78) and (82), we have

Using the same argument as in the proof of Theorem 16, it is easy to derive the desired operator inequalities from the above.
Meanwhile, if , then . Thus, from (78) and the first inequality of (82), we have

Letting , we can obtain the desired operator inequalities.

Competing Interests

The author declares that there is no competing interests regarding the publication of this paper.

References

M. Sababheh and D. Choi, “A complete refinement of Young's inequality,” Journal of Mathematical Analysis and Applications, vol. 440, no. 1, pp. 379–393, 2016.

J. Wu and J. Zhao, “Operator inequalities and reverse inequalities related to the Kittaneh-Manasrah inequalities,” Linear and Multilinear Algebra, vol. 62, no. 7, pp. 884–894, 2014.

W. Liao and J. Wu, “Improved Young and Heinz inequalities with the Kantorovich constant,” Journal of Mathematical Inequalities, vol. 10, no. 2, pp. 559–570, 2016.

T. Ando, “Matrix Young inequalities,” in Operator Theory in Function Spaces and Banach Lattices: Essays Dedicated to A.C. Zaanen on the Occasion of his 80th Birthday, vol. 75 of Operator Theory Advances and Applications, pp. 33–38, Springer, Berlin, Germany, 1995.

R. Bhatia and C. Davis, “More matrix forms of the arithmetic-geometric mean inequality,” SIAM Journal on Matrix Analysis and Applications, vol. 14, no. 1, pp. 132–136, 1993.