Matrix operator inequalities based on Mond, Pecaric, Jensen and Ando s

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1 RESEARCH Matrix operator inequalities based on Mond, Pecaric, Jensen and Ando s Xue Lanlan and Wu Junliang * Open Access * Correspondence: lwu678@tom com College of Mathematics and Statistics of Chongqing University, Chongqing 4033, China Abstract This work is concerned with discussing matrix operator s inequalities based on Mond and Pecaric s matrix inequalities and matrix operator inequalities for convex and concave functions that occur in inequalities of Jensen and Ando s for positive linear maps Keywords: positive definite, convex concave function, operator mean, Hermitian matrix Introduction Operator inequalities play important roles in various mathematical and statistical contexts In early of study operator inequalities, there have been many research results, where useful references centered mainly literatures [-4] In nearly 0 years, some new research results of operator inequalities continuously appears, readers may refer to Mond and Pecaric [5], Beesack [], Hansen [6] and Kubo and Ando [] and so on In [5], Mond Pecaric gave several matrix operator inequalities associated with positive linear maps by means of concavity and convexity theorems Beesack and Pĕcarić in [] showed the Jensen s inequality for real valued convex functions Hansen in [6] also gave operator inequalities associated with Jensen s inequality In this article, we will give some new operator inequalities of matrix version based on Mond, Pecaric, Jensen and Ando s This article is organized as follows: In Section, based on Mond and Pecaric s operator inequalities we give matrix operator inequalities of real valued convex functions In Section 3, we first present some new matrix operator inequalities by means of Jensen s and Ando s matrix inequalities, then we give Jensen inequality for convex functions in matrix form and conduct further research on the properties of Jensen inequality Throughout the article, A =,,,k denote positive definite Hermitian matrices of order n n with eigenvalues in interval [m,m] 0<m <M, U =,,,k denote r n matrices such that U U = I IfAis a n n Hermitian matrix, then there exists a unitary matrix U such that A = U*[l,,l n ]U, where [l,,l n ] is diagonal matrix and l i i =,,,n areeigenvaluesofa, respectively f A is then defined by fa =U*[f l,,fl n ]U A B means A-B is positive semi-definite 0 Lanlan and Junliang; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

2 Page of 3 If Ft =Fft,gt, then we use FA to denote matrix operator of FfA, ga, it means that FA =FfA,gA, while FA,B denotes the matrix function of two variables as it was well defined in some known articles Matrix operator inequalities based on Mond and Pecaric s In [5], Mond and Pecaric once gave two important matrix inequalities which are the following Theorems A and B: Theorem A LetA =,,,k be Hermitian matrices of order n n with eigenvalues in interval [m,m], and let U =,,,k be r n matrices such that U U = I If f is a real valued continuous convex function on [m,m], then U f A U MI k U A U U A U mi f m + f M Theorem B LetA =,,,k be Hermitian matrices of order n n with eigenvalues in interval [m,m], and let U =,,,k be r n matrices those such that U U = I Iffis a continuous convex function on [m,m], J is an interval such that J [m,m] andfu,v is a real valued continuous function defined on J J and it is a matrix increasing function to its first variable, then F U f A U, f U A U { [ M x max F x [m,m] f m + x m ]} f M, f x I { = max F [ θf m + θ f M, f θm + θ M ] } I θ [0,] From Theorems A and B, we can make a more profound study and get more results which have the extensive meaning Theorem Let A =,,,k be Hermitian matrices of order n n with eigenvalues in interval [m,m], and U =,,,k be r n matrices such that U U = I If f is a real valued continuous convex function on [m,m], then U f mi + MI A U U A U mi f m + f m I + f M I U A U MI k U A U f M Proof BytheproofofTheoremof[5],weknowthatforanyrealvaluedconvex function, the following inequality holds

3 Page 3 of 3 f m + M z + M z f m + = z m f m + M z = f m + f M m + M z m f M f M M z f m + z m f M f m + f M f z m + M z [m, M] It follows at once from the above inequality that f mi + MI A f m I + f M I f A, where mi A MI =,,,k The following matrix inequality can be deduced from above one U f mi + MI A U f m I + f M I U f A U Similarly, we can get the following theorem: Theorem Let A =,,,k be Hermitian matrices of order n n with eigenvalues in the interval [m,m], and let U =,,,k be r n matrices such that U U = I Iffis a real valued continuous convex function on [m,m], then U f mi + MI A U m + M f I U f A U m + M MI k U A U U A U mi f I f m + f M Proof By the proof of Theorem, we know that for any a real valued convex function, the following inequality holds: m + M f m + M z f m + M f f z M z f m + z m f M m + M z [m, M] It follows from the above inequality that f m + M mi + MI A f I f A m + M MI A f I f m + A mi f M, where mi A MI =,,,k Further, the following matrix inequality can be deduced from above one

4 Page 4 of 3 U f mi + MI A U m + M f I U f A U m + M MI k U A U U A U mi f I f m + f M As a special case of Theorem B, we give the following: Theorem 3 LetA =,,,k be Hermitian matrices of order n n with eigenvalues in interval [m,m] and U =,,,k be r n matrices such that U U = I Iff is a continuous convex function on [m,m], J is an interval which satisfy J [m,m], Fu,v =u-v is matrix increasing in its first variable, then the following inequality holds: U f A U f U A U [ ] max θf m + θ f M f θm + θ M I θ [0,] We can also get U f A U [ ] θf m + θ f M max f θ [0,] f θm + θ M U A U 3 Matrix operator inequalities based on Jensen and Ando s In this section, we discuss two issues, the first is matrix operator inequalities that get based on an important lemma, and the second is the Jensen inequality of matrix form and further research on the properties of the inequality In the first part, we suppose A is a positive definite Hermitian matrix of order n n with mi A MI, ft is defined as a nonnegative concave function on [m,m] with0 <m <M, thenfa is defined as a positive operator by the usual functional calculus A real valued function f is said to be operator monotone on interval [m,m] ifa B f A fb for every positive definite Hermitian matrices A and B whose eigenvalues are contained on [m,m], and f is said to be operator concave on [m,m] iffla+-lb lfa+-lfb for all real numbers 0 l and every positive definite Hermitian matrices A and B whose eigenvalues are contained in [m,m] We begin with the following lemma Lemma 3 Let A =,,,k be positive definite Hermitian matrices of order n n with eigenvalues in [m,m], ft beanonnegativerealvaluecontinuousstrictlyconcave twice differentiable function on [m,m] with0<m <M, andu =,,,k be r n matrices such that U U = I Then for any given a >0, U f A U αf U A U + βi holds for β = β m, M, f, α = at + b αf t 0,

5 Page 5 of 3 where t 0 is defined as the unique solution of f t = a when α f M a α f f M f m m a =, b = Mf m mf M ; Otherwise t 0 is defined as M or m according as a α f M or f m a α Proof Wetakeht =at+b-aft 0 Since ft is strictly concave, its derivative f t is strictly decreasing Hence, if f M a α f m,thenh t =0occursininterval[m, M] onlyatpointt 0 Sinceh t =-af t > 0, this means that min m t M h t = h t 0 β Next, we consider a α f M Since, h t <0,ht is decreasing on [m,m] Therefore, we can deduce that we deduce that [m,m] min m t M h t = h t 0 β as t0 = M Similarly, if f m a α,then min m t M h t = h t 0 β as t0 = m It follows that at+b aft+b for tî If applying the inequality above to a where mi U A U, we have U A U + bi αf U A U + βi, U A U MI On the other hand, since ft is concave, we have ft at+b for tî[m,m], applying this inequality to A, we have fa aa +bi, and then U f A U a U A U + bi Combining these two inequalities we obtain U f A U αf U A U + βi Use a similar method, the following Theorem can be proved easily Remark 3 If we put a = in Lemma 3, then βi f U A U U f A U holds for b = at+b-ft 0 andt 0 satisfy f t =a Sincef is strictly concave, it can deduce that f M a f m The following corollaries are the special case of Lemma 3

6 Page 6 of 3 Corollary 3 Let,A =,,,k be positive definite Hermitian matrices of order n n with eigenvalues in interval [m,m] that 0 <mi A MI and 0 <p < resp p <0,p >, and let U =,,,k be r n matrices such that U U = I Then for any given a >0 p p U A p U α U A U + βi resp U A p U α U A U + βi holds for β = β m, M, t p, α α p M p m p p p Mm p mm p + αp = if pm p M p m p α pmp, min { α M p, α m p} otherwise, resp β = β m, M, t p, α α p M p m p p p Mm p mm p + αp = if pm p M p m p α pmp, max { α M p, α m p} otherwise Proof We only prove the former, as in proof of Theorem 3, let ft =t p, since t 0 is defined as the unique solution of f t = a α when f m a α f M,soweget t 0 = a p pα and f t 0 = a α = ptp,whenpm p α β = α p αp M p m p M p m p p p Mm p mm p + pmp,wherea = f M f m Mf m mf M b = Otherwise b = min{-am p,-am p } Hence the results obtained by the conclusion of Lemma 3 since ft =t p As a special case of Corollary3, we get the following corollary Corollary 3 Let A =,,,k be positive definite Hermitian matrices of order n n with eigenvalues in the interval [m,m] that 0 <mi A MI, let U =,,,k be r n matrices such that U U = I Then for any given a >0 U A U α U A U + βi by and

7 Page 7 of 3 holds for M + m α β = β m, M, t, α = mm { Mm α max m, a } M if m M α M m, if either 0 <α< m M or M m <α In particular, M m U A U U A U I Mm U A U U A U M + m 4Mm Proof The results obtain by the conclusion of Corollary 3 as ft =t - Remark 3 If we put a = and p = in Corollary 3, then U A U U A U I 4 The results were obtained by Liu and Neudecker [7] The theory of operator means for positive linear operators on a Hilbert space which in connection with Löwner s theory for operator monotone functions was established by Kubo and Ando [8] In the following discussion, as matrix versions, we will give the generalization of matrix inequalities through the theory of operator means which we apply it to derive Jensen s and Ando s matrix inequalities For that we give the following definition: AmapA,B AsB is called an operator mean if the following conditions are satisfied: M Monotonicity: A C and B D imply AsB CsD; M upper continuity: A n A and B n B imply A n sb n AsB; M3 transformer inequality: UAsBU* UAU*sUBU* for any operator U If f is nonnegative operator concave function, we define an operator mean s through the formula Aσ B = A f A BA A for all positive definite Hermitian matrices A and B of order n n Simple example of operator means is the geometric mean # defined by A# B = A A BA A

8 Page 8 of 3 Before proceeding further, we present a useful lemma Lemma 3 Letf be a nonnegative operator concave function on [m,m] with0<m <M, for all positive definite Hermitian matrices A and B, letu be any r n matrix such that UU* = I Then the following statements are mutually equivalent: i UAsBU* UAU*sUBU*, ii UfAU* fuau* Proof By i, ii is obvious as f is concave Then we will show that ii implies i In fact, U Aσ B U = UA f A BA A U = UAU UAU UA f A BA A U UAU UAU = UAU UAU UA f A BA A U UAU UAU UAU f UAU UA A BA A U UAU UAU = UAU f UAU UBU UAU UAU = UAU σ UBU holds for VV = UAU UA A U UAU = I, where V = UAU UA We now introduce some new theorems of operator means for concave matrix functions Theorem 3 LetA and B be positive definite Hermitian matrices such that 0 <m I A M I,0<m I B M I and ft be a nonnegative real valued continuous twice differentiable function on [m,m] with0<m <M, letu be r n matrix such that UU* = I Then UAsBU* auau*subu*+buau* holds for any given a > 0 and b = b m,m,f,a =at+b-aft 0 where t 0 is defined as the unique solution of f t = a α when f M a α f f M f m Mf m mf M m, here a = and b = ; Otherwise t 0 is defined as M or m according as a α f M or f m a α,where m = m M and M = M m UBsAU* aubu*suau*+bubu* holds for which is defined ust as above with m = m, M = M M m Proof We only prove the former It follows from Lemma 3 that for a given a >0 UAU UA f A BA A U UAU αf UAU UA A BA A U UAU + βi

9 Page 9 of 3 holds for VV = UAU UA A U UAU = I and β = β m, M, f, α, M m where V = UAU UA and U Aσ B U = UA f A BA A U = UAU UAU UA f A BA A U UAU UAU = UAU UAU UA f A BA A U UAU UAU UAU αf UAU UA A BA A U UAU + βi UAU = UAU αf UAU UBU UAU + βi UAU = α UAU σ UBU + βuau Remark 33 If we put a = in Theorem 3, then we have the following: βuau UAU σ UBU U Aσ B U resp βubu UBU σ UAU U Bσ A U holds for b = at+b-ft 0 and t 0 such that f t =a, where m = m and M = M M m respm = m, M = M M m If we put in Theorem 3, the following Corollary 33 follows from Theorem 3 f t = t Corollary 33 Let A and B be positive definite Hermitian matrices such that 0 <m I A M I and 0 <m I B M I Let U be r n matrix such that UU* =I Then for any given a >0 U A# B U α UAU # UBU + βuau holds for β = β m, M, t, α Mm mm α M + m = M m 4 if min α M, α m m M + m α otherwise, M M + m,

10 Page 0 of 3 where m = m and M = M, and M m U B# A U α UBU # UAU + βubu holds for b which is defined ust as above with m = m, M = M M m In the following part, according to the traditional Jensen inequality of convex function, we discuss Jensen inequality in matrix form that based on Hermitian matrix and conduct further research on the properties of Jensen inequality An inequality be said to be a Jensen inequality of Hermitian matrix is means that a convex function f defined on I =[m,m] satisfies f P i A i P i f A i,wherea i i =,,k aren n Hermitian and commutative matrices with eigenvalues in [m,m], P i >0i =,,k with P i = Lemma 33 Let f be convex function on I =[m,m], l i i =,,k Î [m,m] and P i >0 i =,,k with P i =, λ i =,, n [m, M], then P i f λ i f P i λ i α f m + β f M f α m + β M { } { } Proof Sinceλ i =,, n [m, M], there are sequences ui, vi [0, ] with u i + v i =, such that λ i = u i m + v i M, =,,, n, hence P i f λ i f P i λ i = P i f u i m + v i M [ f P i ui m + v i M ] = P i f u i m + [ u i M f P i ui m + ] u i M P i ui f m + u i f M f [m ] P i u i + M P i ui = f m P i u i + f M [ P i u i f m P i u i + M ] P i u i Denoting P i u i = α, Consequently, P i u i = β, we have that 0 a,b with a +b = P i f λ i f P i λ i α f m + β f M f α m + β M

11 Page of 3 From this lemma, we have Theorem 3 Letfbe convex function on I =[m,m], A i i =,,k, are Hermitian matrices of order n n with eigenvalues in [m,m] and commutative matrices; P i >0i =,,k with P i =, then [ P i f A i f P i A i max α f m + β f M f α m + β M ] I α Proof As the proof in the definition, by Lemma 3, we get P i f A i f P i A i = P i U diag [ f [ ] λ i,, f λin U f P i U diag ] λ i,, λ in U [ = U diag P i f λ i,, P i f ] [ ] λ in U U diag f P i λ i,, f P i λ in U [ = U diag P i f λ i f P i λ i,, P i f ] λ in f P i λ in U U diag [ α f m + β f M f α m + β M,, α n f m + β n f M f α n m + β n M ] U [ max α f m + β f M f α m + β M ] I α Apply Theorem 3 with f = x, we obtain the following corollary which we call it pre-gruss inequality of Hermitan matrix Corollary 34 Let A i i =,,k, P i be defined as above, then P i A i P i A i 4 I Proof By Theorem 3, we obtain at once the following inequality P i A i P i A i max α m + β M α m + β M I α max α β I α = 4 I For P = P = = P k = k Let us define { P 0 = k, k,, k } k From Theorem 3, we have

12 Page of 3 Corollary 35 Let f, A i i =,,k, P i be defined as above, then f A + f A + + f A k A + A + + A k f k k max p P 0 [ pf m + p f M f pm + p M ] I Theorem 33 Let f, A i i =,,k, P i be defined as above, then we have [ ] M + m P i f A i f P i A i f m I + f M I f I Proof From Theorem 3, we have P i f A i f P i A i Indeed, U diag [ α f m + β f M f α m + β M,, α n f m + β n f M f α n m + β n M ] U α f m + β f M f α m + β M = f m + f M [ f α M + β m + f α m + β M ] [ f m + f M f α M + β m + f α m + β M ] [ ] M + m = f m + f M f Hence, [ P i f A i f P i A i f m I + f M I f M + m ] I Moreover, we get further conclusions Theorem 34 Let f, A i i =,,k, P i be defined as above, and f is also a differentiable convex function, then we have P i f A i f P i A i 4 [ f M f m ] I Proof Since f is convex, we have f α m + β M f m + β f m ; f α m + β M f M + α m M f M Therefore, α f m + β f M f α m + β M [ = α f m f α m + β M ] [ + β f M f α m + β M ] [ α f m f M β f m ] [ + β f M f m α m M f M ] α β m M f M + α β f m = α β [ f M f m ] 4 [ f M f m ]

13 Page 3 of 3 Hence, [ P i f A i f P i A i max α f m + β f M f α m + β M ] I α 4 [ f M f m ] I Thus, the proof is complete Authors contributions WJ carried out Matrix operator theory studies, participated in the conception and design XL conceived of the study, participated in its design and drafting the manuscript All authors read and approved the final manuscript Competing interests The authors declare that they have no competing interests Xue Lanlan is responsible for all the responsibility of the article appeared Received: January 0 Accepted: 7 June 0 Published: 7 June 0 References Beesack, PR, Pĕcarić, JE: On Jensen s inequality for convex functions J Math Anal Appl 0, doi:006/00-47x Kubo, F, Ando, T: Means of positive linear operators Math Ann 46, doi:0007/bf Hansen, F: An operator inequality Math Ann 46, doi:0007/bf Gohberg, I, Lancaster, P, Rodman, L: Matrices and Indefinite Scalar Products, OT8 Birkhäuser, Basel Mond, B, Pećarić, JE: Matrix Inequalities for convex functions J Math Anal Appl 09, doi:0006/ maa Hansen, F: Operator inequalities associated with Jensen s inequality In: Russia TM ed Survey on Classical Inequalities pp Kluwer, Dordrecht Liu, S, Neudecker, H: Several matrix Kantorovich-type inequalities J Math Anal Appl 97, doi:0006/ maa Simić, S: On a global upper bound for Jensen s inequality J Math Anal Appl 343, doi:006/ maa doi:086/09-4x-0-48 Cite this article as: Lanlan and Junliang: Matrix operator inequalities based on Mond, Pecaric, Jensen and Ando s Journal of Inequalities and Applications 0 0:48 Submit your manuscript to a ournal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropencom