
TL;DR
This paper presents new extensions and improvements to Young's inequality, providing elementary proofs, comparisons, and novel inequalities that enhance existing results in the field.
Contribution
It introduces new inequalities that extend and improve upon Dragomir's known results, with elementary proofs and comparative analysis.
Findings
New inequalities extending Dragomir's results
Elementary proofs for existing inequalities
Comparative notes on inequality improvements
Abstract
We focus on the improvements for Young inequality. We give elementary proof for known results by Dragomir, and we give remarkable notes and some comparisons. Finally, we give new inequalities which are extensions and improvements for the inequalities shown by Dragomir.
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Further improvements of Young inequality
Shigeru Furuichi1111E-mail:[email protected]
1Department of Information Science, College of Humanities and Sciences, Nihon University,
3-25-40, Sakurajyousui, Setagaya-ku, Tokyo, 156-8550, Japan
Abstract. We focus on the improvements for Young inequality. We give elementary proof for known results by Dragomir, and we give remarkable notes and some comparisons. Finally, we give new inequalities which are extensions and improvements for the inequalities shown by Dragomir.
**Keywords : ** Young inequality and operator inequality
**2010 Mathematics Subject Classification : ** 26D07, 26D20 and 15A45
1 Introduction
For and , the inequality
[TABLE]
holds and it is called Young inequality. This inequality is simplified as
[TABLE]
for and . We use this simplified notation for some inequalities throughout this paper. Recently, a number of refinements for Young inequality are studied. In this paper, we focus on the refinements for Young inequality by Dragomir. We give alternative proofs of refined Young inequalities given in [1, 2], with elementary calculations. We also show the inequalities we proved in the previous paper [6], give better estimates than ones proved in [2]. Finally we extend and improve the inequalities given by Dragomir in [1, 2]. We also give the inequalities for the operator version as corollaries.
2 Some remarks for recent results
Recently, Dragomir established the following refinement of Young inequality in [1].
Theorem 2.1
([1])* For and ,*
[TABLE]
Proof: To prove the inequality (1), we put
[TABLE]
Then we calculate
[TABLE]
where
[TABLE]
The last inequality is due to Lemma 2.2 in the below. Therefore we have if and if . Thus we have which implies the inequality (1).
∎
Lemma 2.2
For and , we have
[TABLE]
Proof: For any and , . In addition, since , for and , for . Therefore we have the desired inequality.
∎
Remark 2.3
It is known the following inequality (see [4, 5]),
[TABLE]
where is the Kantorovich constant, and . By the numerical computations, we find that there is no ordering between and so that Theorem 2.1 is not trivial one. Actually, we set the function as
[TABLE]
for and . Then we have and . (In addition, we have similarly and .)
Dragomir also established the following refined Young inequalities with the general inequalities in his paper [2].
Theorem 2.4
([2])* Let and .*
- (i)
If , then
[TABLE]
- (ii)
If , then
[TABLE]
Proof:
- (i)
We prove the first inequality of (3). To do this, we set
[TABLE]
Then we calculate
[TABLE]
where
[TABLE]
From Lemma 2.5 in the below, which means . Thus we have which means the first inequality of (3) hold for and .
We prove the second inequality of (3). To do this, we set
[TABLE]
Then we calculate
[TABLE]
where
[TABLE]
Since for , we have , where
[TABLE]
Then we calculate
[TABLE]
for and . Thus we have which implies which means . Thus we have which means the second inequality of (3) hold for and .
- (ii)
We prove the first inequality of (4). The functions , and were defined in the process of the proof of the second inequality in (i). Here we set the function for
[TABLE]
Then we calculate
[TABLE]
Thus we have so that we have . Thus we have for so that which implies . Therefore we have so that which implies the first inequality of (4).
We prove the second inequality of (4). The functions , and were defined in the process of the proof of the first inequality in (i). For , we easily find so that . Thus we have which means . Therefore we have which implies the second inequality of (4).
∎
Lemma 2.5
For and , we have
[TABLE]
Proof: We set the function as
[TABLE]
Then we calculate
[TABLE]
Thus we have so that we have . Therefore we have which implies the inequality (5).
∎
Remark 2.6
The second inequalities (3) and (4) refine the second inequality in [3, Corollary2.2 (i)].
We obtained the following results in our previous paper.
Proposition 2.7
([6])* For and , we have*
[TABLE]
where
[TABLE]
Remark 2.8
As shown in our previous paper [6], we have the inequality
[TABLE]
for and . That is, the second inequality in Proposition 2.7 gives better bound than the inequality (1), in case of and .
In the following proposition, we give the comparison on bounds in (i) Theorem 2.4 and in Proposition 2.7.
Proposition 2.9
For and , we have
[TABLE]
and
[TABLE]
Proof: We use the inequality
[TABLE]
Then we calculate
[TABLE]
for and . Thus the inequality (6) was proved.
Putting , the inequality (7) is equivalent to the inequality
[TABLE]
For the special case or , the equality holds in (8) so that we assume . Then we use the inequality
[TABLE]
We calculate
[TABLE]
where
[TABLE]
We prove . To this end, we calculate
[TABLE]
Thus we have so that we have , where . From Lemma 2.10, for . Thus we have which implies . Therefore we have
[TABLE]
for and . Taking account for the equality cases happen if or , we have the inequality (7).
∎
Lemma 2.10
For , we have
[TABLE]
Proof: Since for , it is sufficient to prove . So we put . Then we have , and . Therefore we have .
∎
Remark 2.11
As for the bounds on the ratio of arithmetic mean to geometric mean , Proposition 2.9 shows Proposition 2.7 is better than (i) of Theorem 2.4, for the case .
3 Further improvement of Young inequality
We give new improvement of Young inequality which is a further improvement of Theorem 2.1. Throughout this section, we use the generalized exponential function defined by for and with under the assumption that .
Theorem 3.1
For and ,
[TABLE]
Proof: We set the function
[TABLE]
for and . Then we have
[TABLE]
Since for and for , . Thus we have when and for , and for . Therefore we have .
∎
Lemma 3.2
The function defined for and , is monotone decreasing in
Proof: We calculate where for . Since , . We thus have .
∎
Corollary 3.3
For , and ,
[TABLE]
Proof: Since , we have the desired result by Theorem 3.1 and Lemma 3.2.
∎
Remark 3.4
Since and Lemma 3.2, we have
[TABLE]
which means that the right hand side in Theorem 3.1 gives the tighter upper bounds of than one in Theorem 2.1.
Remark 3.5
Proposition 2.7 shows the upper bound of is for , while Theorem 3.1 gives the upper bound of for all . In addition, for the case , the right hand side in Theorem 3.1 gives the tighter upper bounds of than in Proposition 2.7.
Remark 3.6
It is easy to see that . As we noted in Remark 2.3, the inequalities (2) are known. By the numerical computations, we have no ordering between and . Actually, we set the function for and as
[TABLE]
Then and .
Remark 3.7
From the proof of Lemma 3.2, we find that the function is monotone decreasing for . However the following inequality does not hold in general
[TABLE]
since we have a counter-example. For example, if we take and , then . This example suggests the optimality of satisfying the inequality (10) is equal to .
As similar way to the above, we can improve Theorem 2.4 in the following.
Theorem 3.8
Let and .
- (i)
If , then
[TABLE]
- (ii)
If , then
[TABLE]
Proof: Firstly, we prove the second inequality in (i). To this end, we set the function as
[TABLE]
for and . Then we find by elementary calculation
[TABLE]
so that . Secondary we prove the first inequality in (i). To this end, we set the function as
[TABLE]
for and . Then we find by elementary calculations
[TABLE]
Thus we have so that which implies .
Thirdly we prove the first inequality in (ii). To this end, we set the function as
[TABLE]
for and . Then we find by elementary calculations
[TABLE]
Since and , we have so that which implies . Thus we have so that .
Finally we prove the second inequality in (ii). To this end, we set the function as
[TABLE]
for and . Since , we have .
∎
Lemma 3.9
The function defined for and , is monotone decreasing in
Proof: We calculate where for . Since , . We thus have .
∎
Corollary 3.10
Let , , and .
- (i)
If , then
[TABLE]
- (ii)
If , then
[TABLE]
Proof: Taking account for , applying two second inequalities in (i) and (ii) of Theorem 3.8 and Lemma 3.2, we obtain two second inequalities in (i) and (ii) of this theorem.
Taking account for and for , for , applying two first inequalities in (i) and (ii) of Theorem 3.8 and Lemma 3.9, we obtain two first inequalities in (i) and (ii) of this theorem.
∎
Remark 3.11
As we noted , and Lemma 3.2 and Lemma 3.9 assure that Theorem 3.8 gives tighter bounds of than Theorem 2.4.
Remark 3.12
Bounds in (i) of Theorem 3.8 can be compared with Proposition 2.7. As for upper bound, gives tighter than the right hand side in the second inequality of (i) of Theorem 3.8. Since for , also gives tighter than the left hand side in the first inequality of (i) of Theorem 3.8. The inequality can be proven in the following. We set , then we have so that .
Note that Remark 3.12 and (i) of Corollary 3.10 give Proposition 2.9. However, Corollary 3.10 gives alternative tight bounds of when .
Remark 3.13
We give comparisons our inequalities obtained in Theorem 3.8 with the inequalities (2). By the numerical computations, we have no ordering between and for . Actually, we set the function for and as
[TABLE]
Then and . (We easily find that .) We also have no ordering between and for . Actually, we set the function for and as
[TABLE]
Then and . (We easily find that .) As for the lower bounds, we have no ordering between and for . Actually, we set the function for and as
[TABLE]
Then and . We also have no ordering between and for . Actually, we set the function for and as
[TABLE]
Then and .
We close this section showing operator versions for Corollary 3.3 and Corollary 3.10 in the following. We denote weighted arithmetic mean and geometric mean for two strictly positive operators and by and , respectively.
Corollary 3.14
Let , and let and be strictly positive operators satisfying (i) or (ii) with and . Then
[TABLE]
where is Kantrovich constant.
Proof: The inequality (10) is equivalent to
[TABLE]
for any . Thus we have the following operator inequality for strictly positive operator such that ,
[TABLE]
Here we put . In the case of (i), we have . Then we have
[TABLE]
In the case of (ii), we also have . Then we also have
[TABLE]
Since is decreasing for , increasing for and , we obtain the desired result by multiplying to both sides in two above inequalities.
∎
Corollary 3.15
Let , , and let and be strictly positive operators satisfying (i) or (ii) with and . Then
[TABLE]
Proof: The inequalities in (i) and (ii) of Corollary 3.10 can be written as
[TABLE]
The rest of the proof goes similar way to the proof of [2, Corollary 1]. We omit its details.
∎
Acknowledgement
The author was partially supported by JSPS KAKENHI Grant Number 16K05257.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] S.S. Dragomir, A note on Young’s inequality, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A. Math., Vol.111(2017), 349–354.
- 2[2] S.S. Dragomir, On new refinements and reverse of Young’s operator inequality, ar Xiv:1510.01314 v 1.
- 3[3] S. Furuichi and N.Minculete, Alternative reverse inequalities for Young’s inequality, Journal of Mathematical Inequalities, Vol.5(2011), 595–600.
- 4[4] H. Zuo, G. Shi, M. Fujii, Refined Young inequality with Kantorovich constant, J. Math. Inequal., Vol.5(2011), 551–556.
- 5[5] W. Liao, J. Wu, J. Zhao, New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math., Vol.(2015), 467–479.
- 6[6] S.Furuichi and H.R.Moradi, Operator inequalities among arithmetic mean, geometric mean and harmonic mean. II, ar Xiv:1705.02185.
