Abstract
In this article, we present exponential-type inequalities for positive linear mappings and Hilbert space operators, by means of convexity and the Mond-Pečarić method. The obtained results refine and generalize some known results. As an application, we present extensions for operator-like geometric and harmonic means inequalities.
1. Introduction
Let be the algebra of bounded linear operators on a complex Hilbert space If is positive, we write Further, we use the notation for the cone of all positive operators in For two self-ajoint operators , we write if . For a real-valued function of a real variable and a self-adjoint operator , the value is understood by means of the functional calculus.
Let be a real interval of any type. A continuous function is said to be operator convex if holds for each and every pair of self-adjoint operators , with spectra in . The notation will be used in the sequel to denote the class of all positive operators satisfying , for some positive scalars and . A linear map is said to be positive if whenever . If, in addition, , it is said to be normalized.
If is operator convex, then, for any normalized positive linear map , we have [1, 2]while we have the reversed inequality if is operator concave, for any self-adjoint operator with spectrum in .
Inequality (1) is not true if is a convex function (rather than operator convex). However, in the interesting paper [3], various complementary inequalities have been presented for convex and concave functions. For example, it is shown that, for any positive number , one can find a constant such thatfor the twice differentiable convex function and any self-adjoint operator on with spectrum in
Earlier, it has been shown that, for any continuous real valued function , one can find positive constants and such that [4]
In Proposition 10, we present a special case of (2) for a particular choice of ; however we present a simple proof for completeness. Then as an application, we present several improvements and extensions of (2) and (3) for a log-convex function .
In the sequel, we adopt the following notations. For a given function , definewhereIf no confusion arises, we will simply write and
Also, for , defineprovided that exists.
It is clear that, for a convex function , one haswhile the inequalities are reversed for a concave function .
Remark 1. Notice that if is convex on an interval containing , then (7) is still valid for any . That is, does not need to be in
Now, if is log-convex, we have the inequality , which simply reads as followswhere the second inequality is due to the arithmetic-geometric inequality. We refer the reader to [5] for some detailed discussion of (8).
Another useful observation about log-convex functions is the following. If is log-convex on and if is such that , (7) implies Simplifying this inequality implies the following.
Lemma 2. Let be log-convex. If is differentiable at , then
In this article, we present several inequalities for log-convex functions based on the Mond-Pečarić method. In particular, we present inequalities that can be viewed as exponential inequalities for log-convex functions. More precisely, we present inequalities among the quantities and
Another interest in this paper is to present inequalities for operator-like means when filtered through normalized positive linear maps. That is, it is known that, for an operator mean and two positive invertible operators , one has [6] In particular, we show complementary inequalities for the geometric and harmonic operator-like means, when Of course, when , these are not operator means. Our results can be considered as extensions of [7, Theorem 2.2].
2. Main Results
Now we proceed to the main results, starting with a complementary result of [8, Corollary 2.5] and [5, Proposition 2.1].
Proposition 3. Let be log-convex, , and be a normalized positive linear map. Thenwhere
Proof. The first and the second inequalities follow from [5, Proposition 2.1] and the fact that . So we have to prove the other inequalities. Applying a standard functional calculus argument for the operator in (8), we getFollowing [8], we have, for , where That is, By setting , we obtain With this choice of , we have , which, together with (16), complete the proof.
Notice that Proposition 3 can be regarded as an operator extension of [9, Theorem 2.5] and a refinement of [8, Corollary 2.5].
Corollary 4. Let and be a normalized positive linear map. Then, for ,where the generalized Kantrovich constant is defined by
Proof. The result follows immediately from Proposition 3, by letting
Remark 5. Corollary 4 presents a refinement of the corresponding result in [10, Lemma 2].
As another application of Proposition 3, we have the following bounds for operator means. To simplify our statement, we will adopt the following notations. For a given function and two positive operators and satisfying , we write
Corollary 6. Let be such that for some positive scalars Then, for any linear map (not necessarily normalized) and any log-convex function ,
Proof. From the assumption , we have Therefore, if is log-convex on , Proposition 3 impliesfor any normalized positive linear map In particular, for the given , define Then, is a normalized linear mapping and the above inequalities imply, upon conjugating with , the desired inequalities.
In particular, Corollary 6 can be utilized to obtain versions for the geometric and harmonic operator means, as follows.
Corollary 7. Let be such that for some positive scalars Then, for any linear map (not necessarily normalized) and for , where and is as in Corollary 4.
Proof. Noting that the function is log-convex on for , the result follows by direct application of Corollary 6.
Remark 8. Recently in [7, Theorem 2.2], the authors proved that if are two positive operators, thenTherefore, Corollary 7 can be regarded as an extension and a reverse for the above inequality, under the assumption with .
Corollary 9. Let be such that for some positive scalars Then, for any linear map (not necessarily normalized) and for , where and
Proof. Noting that the function is log-convex on for , provided that , the result follows by direct application of Corollary 6.
We should remark that the mapping is a decreasing function for In particular,
Further, utilizing (8), we obtain the following. In this result and later in the paper, we adopt the notations:
The following proposition gives a simplified special case of [3, Theorem 2.1].
Proposition 10. Let be convex, , and be a normalized positive linear map. If is either increasing or decreasing on , then, for any ,andprovided that exists and Further, both inequalities are reversed if is concave.
Proof. We give the proof for the reader’s convenience. Notice first that being either increasing or decreasing ensures that Using a standard functional calculus in (7) with and applying to both sides implyOn the other hand, applying the functional calculus argument with impliesNoting that and have the same sign, both desired inequalities follow from (33) and (34).
Now if was concave, replacing with and noting linearity of imply the desired inequalities for a concave function.
As an application, we present the following result, which has been shown in [3, Corollary 2.8].
Corollary 11. Let . Then, for a normalized positive linear mapping ,and
Proof. Let Then is convex and monotone on Letting , direct calculations show that , Then inequality (31) implies the first inequality. The second inequality follows similarly by letting
Manipulating Proposition 10 implies several extensions for log-convex functions, as we shall see next.
We will adopt the following constants in Theorem A.
, , , for and , , , and for
The first two inequalities of the next result should be compared with Proposition 3, where a reverse-type is presented now.
Theorem A. Let be log-convex, , and be a normalized positive linear map. Then, for any ,
Proof. For , we clearly see that is convex and monotone on . Notice that This proves the first two inequalities. Now, for the third inequality, assume that and let Then the second inequality can be viewed asSince , it follows that is operator concave. Therefore, noting (39) and (1), we have which is the desired inequality in the case
Now, if , the function is convex and monotone. Therefore, taking in account (39) and (31), we obtain which completes the proof.
For the same parameters as Theorem A, we have the following comparison too, in which the first two inequalities have been shown in Theorem A.
Corollary 12. Let be log-convex, , and be a normalized positive linear map. Then
Proof. We prove the last inequality. Letting be a normalized positive linear map and noting that is order preserving, the fourth inequality of Proposition 3 implies which is the desired inequality.
For the next result, the following constants will be used.
, , , for and , , , and for
Theorem B. Let be log-convex, , , and be a normalized positive linear map. If , On the other hand, if ,
Proof. Letting and , we have which completes the proof for the case
Now if , we have which completes the proof.
Remark 13. In both Theorems A and B, the constants and can be selected to be 1, as follows. Noting that the function in both theorems is continuous on and differentiable on , the mean value theorem ensures that for some This implies , since we use the notation . A similar argument applies for These values of can be easily found.
Moreover, one can find so that , providing a multiplicative version. Since this is a direct application, we leave the tedious computations to the interested reader.
Utilizing Lemma 2, we obtain the following exponential inequality.
Proposition 14. Let be log-convex, , , and be a normalized positive linear map. Then and where and for
Proof. By Lemma 2, we have A functional calculus argument applied to this inequality with implies which completes the proof of the first inequality. The second inequality follows similarly using (31).
3. Further Refinements
The above results are all based on basic inequalities for convex functions. Therefore, refinements of convex functions inequalities can be used to obtain sharper bounds. We give here some examples. In [11], the following simple inequality was shown for the convex function ,This inequality can be used to obtain refinements of (31) and (32) as follows. First, we note that the function is a continuous function. Further, noting that one can apply a functional calculus argument on (52). With this convention, we will use the notation The following is a refinement of Proposition 10. Since the proof is similar to that of Proposition 10 utilizing (52), we do not include it here.
Proposition 15. Let be convex, , and be a normalized positive linear map. If is either increasing or decreasing on , then, for any ,andprovided that exists and
Notice that applying this refinement to the convex function implies refinements of both inequalities in Corollary 11 as follows.
Corollary 16. Under the assumptions of Corollary 11, we haveand
Remark 17. Inequality (52) has been studied extensively in the literature, where numerous refining terms have been found. We refer the reader to [12, 13], where a comprehensive discussion has been made therein. These refinements then can be used to obtain further refining terms for Proposition 10.
Further, these refinements can be applied to log-convex functions too. This refining approach leads to refinements of most inequalities presented in this article; where convexity was the key idea. We leave the detailed computations to the interested reader.
Data Availability
All data generated or analysed during this study are included in this published article. There is no experimental data in this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work of the first author (Mohammad Sababheh) is supported by a sabbatical leave from Princess Sumaya University for technology. Shigeru Furuichi was partially supported by JSPS KAKENHI Grant Number 16K05257.