Research Article | Open Access

Daeshik Choi, "Multiple-Term Refinements of Young Type Inequalities", *Journal of Mathematics*, vol. 2016, Article ID 4346712, 11 pages, 2016. https://doi.org/10.1155/2016/4346712

# Multiple-Term Refinements of Young Type Inequalities

**Academic Editor:**Shaofang Hong

#### Abstract

Recently, a multiple-term refinement of Youngâ€™s inequality has been proved. In this paper, we show its reverse refinement. Moreover, we will present multiple-term refinements of Youngâ€™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 byAnother 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] showedwhereNote that (7) improves (3), since for all . In [3], Liao and Wu improved (4) as follows:whereThe 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. Thenfor .*

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 hasIf is even, then and by induction. Since , we haveUsing 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 showNote that since at , (19) holds for . We will prove (19) by induction on . Since , one hasAssume that (19) holds and . If is even, then If is odd, thenSince is the line segment joining and on , we haveBy 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 . Sinceone 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 hasThat 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 tofor . Taking the natural logarithm, it suffices to show thatA direct computation shows thatThus for any .

Lemma 11. *For a positive integer and , define byfor . 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 asBy Lemma 11, the above can be expressed byThus it suffices to show that if , thenfor . Replacing by and letting , the above is equivalent toLet . By Lemma 10, one gets that Sinceby Lemma 4, it follows that if , then and therefore (50) holds.

Note that (46) can be written aswhich 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 inwhich is the first inequality of (3), we obtainSince , 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, byIn 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*