# Exponential Inequalities for Positive Linear Mappings

Authors: Mohammad Sababheh; Hamid Reza Moradi; Shigeru Furuichi
Journal: Journal of Function Spaces
 (2018)

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## 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.


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## Body

## 1. Introduction

LetB(H) be the C∗ algebra of bounded linear operators on a complex Hilbert space H. If A∈B(H) is positive, we write A≥0. Further, we use the notation B+(H) for the cone of all positive operators in B(H). For two self-ajoint operators A,B∈B(H), we write A≤B if B-A≥0. For a real-valued function f of a real variable and a self-adjoint operator A∈BH, the value fA is understood by means of the functional calculus.LetJ be a real interval of any type. A continuous function f:J→R is said to be operator convex if f1-vA+vB≤1-vfA+vfB holds for each v∈0,1 and every pair of self-adjoint operators A,B∈BH, with spectra in J. The notation B[m,M]+(H) will be used in the sequel to denote the class of all positive operators A∈B(H) satisfying mI≤A≤MI, for some positive scalars m and M. A linear map φ:B(H)→B(K) is said to be positive if φ(A)≥0 whenever A≥0. If, in addition, φ(I)=I, it is said to be normalized.Iff:J→R is operator convex, then, for any normalized positive linear map φ, we have [1, 2](1)fφA≤φfA,while we have the reversed inequality if f is operator concave, for any self-adjoint operator A with spectrum in J.Inequality (1) is not true if f 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 α, one can find a constant β such that(2)φfA≤αfφA+β,for the twice differentiable convex function f:[m,M]→R and any self-adjoint operator A on H with spectrum in [m,M].Earlier, it has been shown that, for any continuous real valued functionf, one can find positive constants α and β such that [4](3)αφfA≤fφA≤βφfA.In Proposition10, we present a special case of (2) for a particular choice of α; 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 f.In the sequel, we adopt the following notations. For a given functionf:[m,M]→R, define(4)Lm,M,ft=am,M,ft+bm,M,f,where(5)am,M,f=fM-fmM-mandbm,M,f=Mfm-mfMM-m.If no confusion arises, we will simply write a[m,M,f]=af and b[m,M,f]=bf.Also, fort0∈(m,M), define(6)L′t0,ft=ft0+f′t0t-t0,provided that f′(t0) exists.It is clear that, for a convex functionf:[m,M]→R, one has(7)L′t0,ft≤ft≤Lm,M,ft,while the inequalities are reversed for a concave function f.Remark 1.
Notice that iff is convex on an interval J containing [m,M], then (7) is still valid for any t0∈J. That is, t0 does not need to be in (m,M).Now, iff:[m,M]→R+ is log-convex, we have the inequality logf(t)≤L[m,M,logf](t), which simply reads as follows(8)ft≤ft-mMfM-tm1/M-m≤Lm,M,ft,m≤t≤M,where 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. Iff is log-convex on [m,M] and if t0∈(m,M) is such that f(t0)≠0, (7) implies (9)L′t0,gt≤g,whereg=logf.Simplifying this inequality implies the following.Lemma 2.
Letf:[m,M]→R+ be log-convex. If f is differentiable at t0∈(m,M), then (10)ft≥ft0expf′t0ft0t-t0,m≤t≤M.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(11)φfA,fφA,φfA-mMfM-Am1/M-mand (12)fφA-mMfM-φAm1/M-m.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 meanσ and two positive invertible operators A,B, one has [6] (13)φAσB≤φAσφB.In particular, we show complementary inequalities for the geometric ♯t and harmonic !t operator-like means, when t<0. Of course, when t<0, 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.
Letf:[m,M]→R+ be log-convex, A∈B[m,M]+(H), and φ be a normalized positive linear map. Then(14)1μm,M,fφfA≤1μm,M,fφfA-mMfM-Am1/M-m≤fφA≤fφA-mMfM-φAm1/M-m≤μm,M,fφfAwhere(15)μm,M,f≡max1ftM-tM-mfm+t-mM-mfM:m≤t≤M.Proof.
The first and the second inequalities follow from [5, Proposition 2.1] and the fact that μm,M,f>0. So we have to prove the other inequalities. Applying a standard functional calculus argument for the operator φA in (8), we get(16)fφA≤fφA-mMfM-φAm1/M-m≤Lm,M,fφA.
Following [8], we have, for α>0, (17)Lm,M,fφA-αφfA=afφA+bf-αφfA≤β,where β=maxt∈[m,M]{aft+bf-αf(t)}. That is, (18)Lm,M,fφA≤αφfA+β.By setting β=0, we obtain α=maxt∈[m,M]aft+bf/f(t)≔μ(m,M,f). With this choice of α, we have Lm,M,fφA≤μ(m,M,f)φ(f(A)), which, together with (16), complete the proof.Notice that Proposition3 can be regarded as an operator extension of [9, Theorem 2.5] and a refinement of [8, Corollary 2.5].Corollary 4.
LetA∈B[m,M]+(H) and φ be a normalized positive linear map. Then, for t<0,(19)1Km,M,tφAt≤1Km,M,tφMA-mmM-At/M-m≤φAt≤MφA-mmM-φAt/M-m≤Km,M,tφAtwhere the generalized Kantrovich constant is defined by (20)Km,M,t=mMt-Mmtt-1M-mt-1tMt-mtmMt-Mmtt.Proof.
The result follows immediately from Proposition3, by letting f(x)=xt.Remark 5.
Corollary4 presents a refinement of the corresponding result in [10, Lemma 2].As another application of Proposition3, we have the following bounds for operator means. To simplify our statement, we will adopt the following notations. For a given function f:[m,M]→[0,∞) and two positive operators A and B satisfying mA≤B≤MA, we write (21)AσfB≔A1/2fA-1/2BA-1/2A-1/2andAδB=A-1/2BA-1/2.Corollary 6.
LetA,B∈B+(H) be such that mA≤B≤MA for some positive scalars m,M. Then, for any linear map φ (not necessarily normalized) and any log-convex function f:[m,M]→R+, (22)1μm,M,fφAσfB≤1μm,M,fφA1/2fAδB-mMfM-AδBm1/M-mA1/2≤φAσfφB≤φA1/2fφAδφB-mMfM-φAδφBm1/M-mφA1/2≤μm,M,fφAσfB.Proof.
From the assumptionmA≤B≤MA, we have m≤AδB≔A-1/2BA-1/2≤M. Therefore, if f is log-convex on [m,M], Proposition 3 implies(23)1μm,M,fψfAδB≤1μm,M,fψfAδB-mMfM-AδBm1/M-m≤fψAδB≤fψAδB-mMfM-ψAδBm1/M-m≤μm,M,fψfAδB,for any normalized positive linear map ψ. In particular, for the given φ, define (24)ψX=φA-1/2φA1/2XA1/2φA-1/2.Then, ψ is a normalized linear mapping and the above inequalities imply, upon conjugating with φ(A)1/2, the desired inequalities.In particular, Corollary6 can be utilized to obtain versions for the geometric and harmonic operator means, as follows.Corollary 7.
LetA,B∈B+(H) be such that mA≤B≤MA for some positive scalars m,M. Then, for any linear map φ (not necessarily normalized) and for t<0, (25)1Km,M,tφA♯tB≤1Km,M,tφAσgB≤φA♯tφB≤φAσgφB≤Km,M,tφA♯tB,where g(x)=Mx-mmM-xt/(M-m) and K(m,M,t) is as in Corollary 4.Proof.
Noting that the functionf(x)=xt is log-convex on [m,M] for t<0, the result follows by direct application of Corollary 6.Remark 8.
Recently in [7, Theorem 2.2], the authors proved that if A,B are two positive operators, then(26)φA♯tφB≤φA♯tBt∈-1,0.Therefore, Corollary 7 can be regarded as an extension and a reverse for the above inequality, under the assumption mA≤B≤MA with M≥m>0.Corollary 9.
LetA,B∈B+(H) be such that mA≤B≤MA for some positive scalars m,M≥1. Then, for any linear map φ (not necessarily normalized) and for t<0, (27)1Hm,M,tφA!tB≤1Hm,M,tφAσgB≤φA!tφB≤φAσgφB≤Hm,M,tφA!tB,where g(x)=1!tMx-m1!tmM-x1/(M-m) and (28)Hm,M,t=1-t2+tmM21-tmM+t1!tm1!tM.Proof.
Noting that the functionf(x)=(1-t+tx-1)-1 is log-convex on [m,M] for t<0, provided that m≥1, the result follows by direct application of Corollary 6.We should remark that the mappingt↦H(m,M,t) is a decreasing function for t<0. In particular, (29)Hm,M,0=1≤Hm,M,t≤mM-12m-1M-1=limt→-∞Hm,M,t,∀t<0.Further, utilizing (8), we obtain the following. In this result and later in the paper, we adopt the notations: (30)αf,t0=aff′t0andβf,t0=aft0+bf-afft0f′t0.The following proposition gives a simplified special case of [3, Theorem 2.1].Proposition 10.
Letf:[m,M]→R+ be convex, A∈B[m,M]+(H), and φ be a normalized positive linear map. If f is either increasing or decreasing on [m,M], then, for any t0∈(m,M),(31)φfA≤αf,t0fφA+βf,t0,and(32)fφA≤αf,t0φfA+βf,t0,provided that f′(t0) exists and f′(t0)≠0. Further, both inequalities are reversed if f is concave.Proof.
We give the proof for the reader’s convenience. Notice first thatf being either increasing or decreasing ensures that aff′(t0)>0. Using a standard functional calculus in (7) with t=A and applying φ to both sides imply(33)ft0+f′t0φA-t0≤φfA≤afφA+bf.On the other hand, applying the functional calculus argument with t=φ(A) implies(34)ft0+f′t0φA-t0≤fφA≤afφA+bf.Noting that af and f′(t0) have the same sign, both desired inequalities follow from (33) and (34).
Now iff was concave, replacing f with -f and noting linearity of φ 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.
LetA∈B[m,M]+(H). Then, for a normalized positive linear mapping φ,(35)φA-1≤φA-1+1m-1M2and(36)φA-1≤M+m24mMφA-1.Proof.
Letf(t)=t-1. Then f is convex and monotone on [m,M]. Letting t0=mM∈(m,M), direct calculations show that α(f,t0)=1, β(f,t0)=1/m-1/M2. Then inequality (31) implies the first inequality. The second inequality follows similarly by letting t0=(m+M)/2.Manipulating Proposition10 implies several extensions for log-convex functions, as we shall see next.We will adopt the following constants in Theorem A.a h = a [ m , M , h ], bh=b[m,M,h], α=α(h,t0), β=β(h,t0) for h(t)=ft-m(M)fM-t(m)1/(M-m) and ah1=a[fM-m(m),fM-m(M),h1], bh1=b[fM-m(m),fM-m(M),h1], α1=α(h1,t1), and β1=β(h1,t1) for h1(t)=t1/(M-m).The first two inequalities of the next result should be compared with Proposition3, where a reverse-type is presented now.Theorem A.
Letf:[m,M]→R+ be log-convex, A∈B[m,M]+(H), and φ be a normalized positive linear map. Then, for any t0,t1∈(m,M), (37)fφA≤fφA-mMfM-φAm1/M-m≤αφfA-m/M-mMfM-A/M-mm+β≤αφfA-mMfM-Am1/M-m+β,M-m≥1αα1φfA-mMfM-Am1/M-m+αβ1+β,M-m<1.Proof.
Forh(t)=ft-m(M)fM-t(m)1/(M-m), we clearly see that h is convex and monotone on [m,M]. Notice that (38)fφA≤fφA-mMfM-φAm1/M-mbythethirdinequalityofProposition3=hφA≤αh,t0φhA+βh,t0by32.=αh,t0φfA-m/M-mMfM-A/M-mm+βh,t0.This proves the first two inequalities. Now, for the third inequality, assume that M-m≥1 and let h1(t)=t1/(M-m). Then the second inequality can be viewed as(39)fφA≤αφh1fA-mMfM-Am+β.Since M-m≥1, it follows that h1 is operator concave. Therefore, noting (39) and (1), we have (40)fφA≤αφh1fA-mMfM-Am+β≤αh1φfA-mMfM-Am+β,which is the desired inequality in the case M-m≥1.
Now, ifM-m<1, the function h1 is convex and monotone. Therefore, taking in account (39) and (31), we obtain (41)fφA≤αφh1fA-mMfM-Am+β≤αα1h1φfA-mMfM-Am+β1+β=αα1φfA-mMfM-Am1/M-m+αβ1+β,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.
Letf:[m,M]→R+ be log-convex, A∈B[m,M]+(H), and φ be a normalized positive linear map. Then (42)fφA≤fφA-mMfM-φAm1/M-m≤αφfA-m/M-mMfM-A/M-mm+β≤αμm,M,fφfA+β.Proof.
We prove the last inequality. Lettingψ(X)=X be a normalized positive linear map and noting that φ is order preserving, the fourth inequality of Proposition 3 implies (43)αφfA-m/M-mMfM-A/M-mm+β=αφfψA-m/M-mMfM-ψA/M-mm+β≤αμm,M,fφψfA+β=αμm,M,fφfA+β,which is the desired inequality.For the next result, the following constants will be used.a ^ h = a [ m , M , h ^ ], b^h=b[m,M,h^], α^=α^(h^,t0), β^=β^(h^,t0) for h^(t)=ft-m(M)fM-t(m) and ah1=a[fM-m(m),fM-m(M),h1], bh1=b[fM-m(m),fM-m(M),h1], α1=α(h1,t1), and β1=β(h1,t1) for h1(t)=t1/(M-m).Theorem B.
Letf:[m,M]→R+ be log-convex, t0,t1∈(m,M), A∈B[m,M]+(H), and φ be a normalized positive linear map. If M-m≥1, (44)φfA≤φfA-mMfM-Am1M-m≤α^fφA-mMfM-φAm+β^1M-m.
On the other hand, ifM-m<1, (45)φfA≤α1φfM-AmfA-mM1/M-m+β1≤α1α^fφA-mMfM-φAm+β^1/M-m+β1.Proof.
Lettingh1(t)=t1/(M-m) and h^(t)=ft-m(M)fM-t(m), we have (46)φfA≤φh1fA-mMfM-AmbythefirstinequalityofProposition3≤h1φfA-mMfM-Amsinceh1isoperatorconcave=φfA-mMfM-Am1/M-m=φh^A1/M-mwhereh^t=ft-mMfM-tm≤α^h^φA+β1/M-mby31=α^fφA-mMfM-φAm+β^1/M-m,which completes the proof for the case M-m≥1.
Now ifM-m<1, we have (47)φfA≤φh1fA-mMfM-AmbythefirstinequalityofProposition3≤α1h1φfA-mMfM-Am+β1by31=α1φfA-mMfM-Am1/M-m+β1=α1φh^A1/M-m+β1≤α1α^h^φA+β^1/M-m+β1by31=α1α^fφA-mMfM-φAm+β^1/M-m+β1,which completes the proof.Remark 13.
In both Theorems A and B, the constantsα and α1 can be selected to be 1, as follows. Noting that the function h in both theorems is continuous on [m,M] and differentiable on (m,M), the mean value theorem ensures that ah=h′(t0) for some t0∈(m,M). This implies α=1, since we use the notation α≡α(h,t0)=ah/h′(t0). A similar argument applies for h1. These values of t0 can be easily found.
Moreover, one can findt0 so that β(h,t0)=0, providing a multiplicative version. Since this is a direct application, we leave the tedious computations to the interested reader.Utilizing Lemma2, we obtain the following exponential inequality.Proposition 14.
Letf:[m,M]→R+ be log-convex, t0,t1∈(m,M), A∈B[m,M]+(H), and φ be a normalized positive linear map. Then (48)φfA≥ft0αexpf′t0ft0φA-t0-βαft0,and (49)fφA≥ft0αφexpf′t0ft0A-t0-βαft0,where α=α(k,t1) and β=β(k,t1) for k(t)=expf′(t0)/f(t0)(t-t0).Proof.
By Lemma2, we have (50)ft≥ft0expf′t0ft0t-t0,m≤t≤M.A functional calculus argument applied to this inequality with t=A implies (51)φfA≥ft0φexpf′t0ft0A-t0=ft0φkA≥ft0kφA-βαby32=ft0αexpf′t0ft0φA-t0-βαft0,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 f:[m,M]→R,(52)ft+2mint-m,M-tM-mfm+fM2-fm+M2≤Lm,M,ft.This inequality can be used to obtain refinements of (31) and (32) as follows. First, we note that the function t↦tmin≔2min{t-m,M-t}/M-m(f(m)+f(M))/2-f(m+M)/2 is a continuous function. Further, noting that (53)mint-m,M-t=M-m+M+m-2t2,one can apply a functional calculus argument on (52). With this convention, we will use the notation (54)Amin≔1M-mfm+fM2-fm+M2M-m+M+m-A.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.
Letf:[m,M]→R+ be convex, A∈B[m,M]+(H), and φ be a normalized positive linear map. If f is either increasing or decreasing on [m,M], then, for any t0∈(m,M),(55)φfA+φAmin≤αf,t0fφA+βf,t0,and(56)fφA+φAmin≤αf,t0φfA+βf,t0,provided that f′(t0) exists and f′t0≠0.Notice that applying this refinement to the convex functionf(t)=t-1 implies refinements of both inequalities in Corollary 11 as follows.Corollary 16.
Under the assumptions of Corollary11, we have(57)φA-1+φAmin≤φA-1+1m-1M2and(58)φA-1+φAmin≤M+m24mMφA-1.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.



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