Multiple-Term Refinements of Young Type Inequalities

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.


Introduction
The classical Young inequality states that if ,  > 0 and 0 ≤ V ≤ 1, then For 0 ≤ V ≤ 1, we define three functions  0 (V),  1 (V), and  0 (V) 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  1 (, ) ≥ 1 for all ,  > 0.
Throughout the paper, we will use the following functions.
Definition 2. For ,  ∈ N and ,  > 0, 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 ,  > 0 and  be any positive integer.Then 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  = 1 and 2, 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.

Multiple-Term Reverse of Young Inequality
From now on, we will fix ,  > 0 and use the following functions: for V ∈ [0, 1] and an integer  > 0. As we will see (Lemma 4),   (/2  ) = 0 for any integer  with 0 ≤  ≤ 2 We can express  0 (V) and  1 (V) as multipart functions as follows: For any  ≥ 0, we can formulate   (V) explicitly.
Lemma 7. Let (V) be the reflection of (V) about the point

Journal of Mathematics
Then each of the following is true.
Corollary 9.For any integer  ≥ 1 and 0 ≤ V ≤ 1, one has Proof.Replacing  and  by their squares in Theorem 8, we obtain that for all V, we derive from the above that Hence (32) follows from the identity (35)

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 0 ≤ V ≤ 1, one has Proof.Replacing  −1  by , the inequality is equivalent to for  > 0. Taking the natural logarithm, it suffices to show that A direct computation shows that Thus () ≥ (1) = 0 for any  > 0.
The following shows a multiple-term refinement of Young inequality involving Kantorovich constants.
Note that (46) can be written as which gives the first inequalities of ( 7) and ( 9) with  = 1 and 2, respectively.Now we consider a reverse inequality corresponding to Theorem 12.A given inequality of the form (1 − V) + V ≥ (V, , ) can be utilized to derive its reverse in many cases.For example, replacing V by 1 − V in which is the first inequality of (3), we obtain Since 2 √  ≤  V  1−V +  1−V  V , the above implies the second inequality of (3).Similarly, replacing V by 1 − V in Journal of Mathematics 7 which is the first inequality of (4), we get Since 2 √  ≤  V  1−V +  1−V  V , 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 ,  > 0, 0 ≤ V ≤ 1, and  ∈ N, one has Proof.Replacing V by 1 − V and  by 2  −  + 1 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  = 1 and 2, respectively.

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.
For ,  > 0 and 0 ≤ V ≤ 1, the V-arithmetic and V-geometric means of  and  are defined, respectively, by In the case V = 1/2, we will omit the V-value in them.For example, ∇ denotes ∇ 1/2 .
The operator version of (1) is well known as follows: for  and  positive invertible operators and 0 ≤ V ≤ 1 (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 () ≥ 0 for every  in the spectrum of  implies that () is a positive operator.
Theorem 14.Let ,  > 0 and 0 ≤ V ≤ 1.Then Proof.For any  > 0, 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 ♯ V  = ♯ 1−V , we can express  , (, ) and  , (, ) by Letting  =  −1/2  −1/2 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.