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 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
Proof. For any , we have by Theorem 8. Thus, for any positive operator , we haveby 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 haveSincewe 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 havefor any . Since is an increasing function on and for , for or . Thus, from (78), we havefor or . Replacing by which satisfies or , we get thatMultiplying 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 havefor 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 , thenIf 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.