DINH, TRUNG HOA

DU, THI HOA BINH

HO, MINH TOAN

Kyoto University Research Information Repository

Title ON SOME INEQUALITIES WITH MATRIX MEANS(Operator monotone functions and related topics)Author(s)DINH, TRUNG HOA; DU, THI HOA BINH; HO, MINHTOANCitation数理解析研究所講究録 (2014), 1893: 67-71Issue Date2014-05URL http://hdl.handle.net/2433/195823RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityON SOME INEQUALITIES WITH MATRIX MEANSDINH TRUNG HOA, DU THI HOA BINH, AND HO MINH TOANABSTRACT. Let $0<m\leq A,$ $B\leq M$ and $\sigma,$ $\tau$ two arbitrary meansbetween harmonic and arithmetic means. Then for every positiveunital linear map $\Phi,$$\Phi(A\sigma B)\leq K(h)\Phi(A\tau B)$ ,$\Phi(A\sigma B)\leq K(h)(\Phi(A)\tau\Phi(B))$ ,$\Phi(A)\sigma\Phi(B)\leq K(h)\Phi(A\tau B)$ ,and$\Phi(A)\sigma\Phi(B)\leq K(h)\Phi(A)\tau\Phi(B)$ ,where $K(h)= \frac{(h+1)^{2}}{4h}$ with $h= \frac{M}{m}$ is the Kantorovich constant.1. INTRODUCTIONThe axiomatic theory for connections and means for pairs of positivematrices have been studied by Kybo and Ando [4]. $A$ binary operation$\sigma$ define on the set of positive definite matrices is called a conncctionif(i) $A\leq C,$ $B\leq D$ implies $A\sigma B\leq B\sigma D$ ;(ii) $C(A\sigma B)C\leq(CAC)\sigma(CBC)$ ;(iii) $A_{n}\downarrow A$ and $B_{n}\downarrow B$ imply $A_{n}\sigma B_{n}\downarrow A\sigma B.$If $I\sigma I=I$ , then $\sigma$ is called a mean.Many authors study matrix inequalities containing means and linearunital positive maps on matrix algebras. Such inequalities are interest-ing by themselves and have many applications in quantum informationtheory.In [2], Lin proved the following Theorem.Theorem 1.1 ([2]). Let $0<m\leq A,$ $B\leq M$ . Then for every positiveunital linear map $\Phi,$(1) $\Phi^{2}(A\nabla B)\leq K^{2}(h)\Phi^{2}(A\# B)$ ,and(2) $\Phi^{2}(A\nabla B)\leq K^{2}(h)(\Phi(A)\#\Phi(B))^{2},$where $K(h)= \frac{(h+1)^{2}}{4h}$ with $h= \frac{M}{m}$ is the Kantorovich constant.2000 Mathematics Subject Classification. $26D15.$Key words and phrases. matrix means, inequalities.1893 2014 67-71 67DINH TRUNG HOA, DU THI HOA BINH, AND HO MINH TOANIt is well-known that the arithmetic mean $\nabla$ is the biggest amongsymmetric means (see [4]). $A$ natural question is that is the theoremabove still true if we replace the biggest means by a smaller one? Inthis note, we consider such inequalities for two different means withKantorovich constant. In applications, we give an analogous result ofUchiyama and Yamazaki in [7].This note is based on preprint [1].2. MAIN RESULTSLemma 2.1. Let $0<m\leq A,$ $B\leq M$ and $\sigma,$ $\tau$ two arbitrary meansbetween harmonic and arithmetic means. Then for evew positive unitallinear map $\Phi,$(3) $\Phi(A\sigma B)+Mm\Phi^{-1}(A\tau B)\leq M+m,$and(4) $\Phi(A)\sigma\Phi(B)+Mm\Phi^{-1}(A\tau B)\leq M+m.$Proof. It is easy to see that$(M-A)(m-A)A^{-1}\leq 0,$or$mMA^{-1}+A\leq M+m.$Consequently,$\Phi(A)+mM\Phi(A^{-1})\leq M+m.$Similarly,$\Phi(B)+mM\Phi(B^{-1})\leq M+m.$Summing up two above inequalities, we get$\Phi(A_{\nabla}B)+mM\Phi((A!B)^{-1})\leq M+m.$Besides, from the general theory of matrix means we know that $\nabla\geq\sigma$and $\tau\geq!$ . Hence,$\Phi(A\sigma B)+mM\Phi^{-1}(A\tau B)\leq\Phi(A\sigma B)+mM\Phi((A\tau B)^{-1})$$\leq\Phi(A\nabla B)+mM\Phi((A!B)^{-1})$$\leq M+m.$By a similar argument, we can get inequality (4) with using the factthat$\Phi(A)\sigma\Phi(B)\leq\Phi(A)_{\nabla}\Phi(B)=\Phi(A_{\nabla}B)$ .$\square$The following Proposition is a generalization of Lin’s result men-tioned in Introduction.68ON SOME INEQUALITIES WITH MATRIX MEANSProposition 2.1. Let $0<m\leq A,$ $B\leq M$ and $\sigma,$ $\tau$ two arbitrarymeans between harmonic and arithmetic means. Then for every positiveunital linear map $\Phi,$(5) $\Phi^{2}(A\sigma B)\leq K^{2}(h)\Phi^{2}(A\tau B)$ ,(6) $\Phi^{2}(A\sigma B)\leq K^{2}(h)(\Phi(A)\tau\Phi(B))^{2}$(7) $(\Phi(A)\sigma\Phi(B))^{2}\leq K^{2}(h)\Phi^{2}(A\tau B)$ ,and(8) $(\Phi(A)\sigma\Phi(B))^{2}\leq K^{2}(h)(\Phi(A)\tau\Phi(B))^{2},$where $K(h)= \frac{(h+1)^{2}}{4h}$ with $h= \frac{M}{m}$ is the Kantorovich constant.Proof. We prove (2.1). The inequality (2.1) is equivalent to the follow-ing$\Phi^{-1}(A\tau B)\Phi^{2}(A\sigma B)\Phi^{-1}(A\tau B)\leq K^{2}(h)$ ,or$||\Phi(A\sigma B)\Phi^{-1}(A\tau B)||\leq K(h)$ .On the other hand, it is well known that [5, Theorem 1] for $A,$ $B\geq 0,$$||AB|| \leq\frac{1}{4}||A+B||^{2}$So, it is necessary to prove that$\frac{1}{4mM}||\Phi(A\sigma B)+mM\Phi^{-1}(A\tau B)||^{2}\leq\frac{(M+m)^{2}}{4Mm},$or,$||\Phi(A\sigma B)+mM\Phi^{-1}(A\tau B)||\leq M+m.$The last inequality follows from Lemma 2.1.Remain inequalities in Proposition can be proved analogously. $\square$Remark 1. As we mentioned in the proof of Proposition 2.1 that forany positive matrices $A,$ $B,$ $\Phi(A\sigma B)\leq\Phi(A\nabla B)$ . From that, it canrise a wrong intuition that the proof of Proposition 2.1 can be obtainedeasily from Theorem 1.1. Unfortunately, the last inequality could notbe squared as it was shown in [2, Proposition 1.2].Theorem 2.1. Let $0<m\leq A,$ $B\leq M$ and $\sigma,$ $\tau$ are two arbitrarysymmetric means. Then for every positive unital linear map $\Phi,$$\Phi(A\sigma B)\leq K(h)\Phi(A\tau B)$ ,$\Phi(A\sigma B)\leq K(h)(\Phi(A)\tau\Phi(B))$ ,$\Phi(A)\sigma\Phi(B)\leq K(h)\Phi(A\tau B)$ ,and$\Phi(A)\sigma\Phi(B)\leq K(h)\Phi(A)\tau\Phi(B)$ ,where $K(h)= \frac{(h+1)^{2}}{4h}$ with $h= \frac{M}{m}$ is the Kantorovich constant.69DINH TRUNG HOA, DU THI HOA BINH, AND HO MINH TOANProof. The proof follows from Proposition 2.1 and the fact that thefunction $f(t)=t^{1/2}$ is operator monotone on $[0, \infty)$ . $\square$Corrolary 2.1. Let $f,$ $g$ be symmetric operator monotone functions on$[0, \infty)$ . Then for any pair $0<m<M,$(9) $\max\{\frac{f(t)}{g(t)}, \frac{f(t)}{g(t)}\}\leq K(h)=\frac{(m+M)^{2}}{4mM}, t\in[m, M].$Proof. It is necessary to apply above Theorem for the symmetric matrixmeans $\sigma$ and $\delta$ corresponding to the functions $f$ and $g$ , and definitionof matrix means via it representation functions. $\square$Inequality (9) is interesting by itself, and the authors do not knowan elementary proof even in the case when $f(t)=\sqrt{t}.$As an application, now we give a similar result as in [7]. Uchiyamaand Yamazaki showed that for an operator monotone function $f$ on$[0, \infty)$ if $f(\lambda B+I)^{-1}\# f(\lambda A+I)\leq I$ for all sufficiently small $\lambda>0,$then $f(\lambda A+I)\leq f(\lambda B+I)$ and $A\leq B$ . By applying Theorem 2.1 weget a similar result for any symmetric means.Corrolary 2.2. Let $f$ be operator monotone function on $[0, \infty)$ and$\sigma$ an arbitrary mean between harmonic and arithmetic ones. If for agiven pair of positive invertible matrices $A,$ $B,$$f(\lambda B+I)^{-1}\sigma f(\lambda A+I)\leq K$for all sufficiently small $\lambda>0$ (where $K$ is Kantorovich constant), then$f(\lambda A+I)\leq f(\lambda B+I)$ and $A\leq B.$REFERENCES[1] Dinh bung Hoa, Du Thi Hoa Binh, Ho Minh Toan, On some inequalities withmatrix means. Preprint.[2] M. Lin. Squaring a reverse $AM$-GM inequality. Studia Math. 215 (2013), 187-194.[3] M. Lin, On an operator Kantorovich inequality for positive linear maps. J. Math.Anal. Appl. 402 (2013) 127-132.[4] F. Kubo, T. Ando, Means of positive linear operators. Math. Ann. 246(1979/80), no. 3, 205-224.[5] R. Bhatia, F. Kittaneh, Notes on matrix arithmeticgeometric mean inequalities.Linear Algebra Appl. 308 (2000) 203-211.[6] R. Kaura, M. Singh, J. S. Aujla, Generalized matrix version of reverse Holderinequality. Linear Algebra and its Applications. 434 (2011) 636-640.[7] M. Uchiyama, T. Yamazaki, A converse of Loewner-Heinz inequality and appli-cations to operator means. Journal of Mathematical Analysis and Applications.413(1) (2014) 422429.70ON SOME INEQUALITIES WITH MATRIX MEANSDUY TAN UNIVERSITY, $K7/25$ QUANG TRUNG, DANANG, VIETNAM$E$-mail address: trunghoa.math@gmail.comHA TAY COLLEGE OF PEDAGOGY, HA NOI, VIETNAM$E$-mail address: hoabinhcdsp@gmail.comINSTITUTE OF MATHEMATICS$E$-mail address: hmtoan@math.ac.vn71

ON SOME INEQUALITIES WITH MATRIX MEANS (Operator monotone functions and related topics)

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ON SOME INEQUALITIES WITH MATRIX MEANS (Operator monotone functions and related topics)

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