Refinements of the operator Jensen-Mercer inequality

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1 Electronic Journal of Linear Algebra Volume 6 Volume 6 13 Article 5 13 Refinements of the operator Jensen-Mercer inequality Mohsen Kian moslehian@um.ac.ir Mohammad Sal Moslehian Follow this and additional works at: Recommended Citation Kian, Mohsen and Moslehian, Mohammad Sal. 13, "Refinements of the operator Jensen-Mercer inequality", Electronic Journal of Linear Algebra, Volume 6. DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact scholcom@uwyo.edu.

2 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October 13 REFINEMENTS OF THE OPERATOR JENSEN MERCER INEQUALITY MOHSEN KIAN AND MOHAMMAD SAL MOSLEHIAN Abstract. A Hermite Hadamard Mercer type inequality is presented and then generalized to Hilbert space operators. It is shown that f M +m n x ia i fm+fm n fx ia i, where f is a convex function on an interval [m,m] containing, x i [m,m], i = 1,...,n, and A i are positive operators acting on a finite dimensional Hilbert space whose sum is equal to the identity operator. A Jensen Mercer operator type inequality for separately operator convex functions is also presented. Key words. Jensen Mercer inequality, Operator convex, Jensen inequality, Hermite Hadamard inequality, Jointly operator convex. AMS subject classifications. 47A63, 47A Introduction. The well-known Jensen inequality for the convex functions states that if f is a convex function on an interval [m,m], then n 1.1 f λ i x i λ i fx i for all x i [m,m] and all λ i [,1] i = 1,...,n with n λ i = 1. The Hermite Hadamard inequality asserts that if f is a convex function on [m,m], then m+m f 1 M m M m fxdx fm+fm. For more information, see [5, 15] and references therein. Mercer [1] established a variant of the Jensen inequality 1.1 as follows. Theorem 1.1. If f is a convex function on [m,m], then 1. f M +m λ i x i fm+fm λ i fx i Received by the editors on December, 1. Accepted for publication on October 6, 13. Handling Editor: Harm Bart. Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, PO Box 1339, Bojnord 94531, Iran; Tusi Mathematical Research Group TMRG, PO Box 1113, Mashhad 91775, Iran kian@member.ams.org, kian@ub.ac.ir. Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures CEAAS, Ferdowsi University of Mashhad, PO Box 1159, Mashhad 91775, Iran moslehian@um.ac.ir, moslehian@member.ams.org. 74

3 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October 13 Operator Jensen Mercer inequality 743 for all x i [m,m] and λ i [,1] i = 1,...,n with n λ i = 1. Inequality 1. is known as the Jensen Mercer inequality. Recently, inequality 1. has been generalized; see [1,, 16]. Let BH be the algebra of all bounded linear operators on a Hilbert space H and I denote the identity operator. An operator A is positive denoted by A if Ax,x for all vectors x H. If, in addition, A is invertible, then it is strictly positive denoted by A >. By A B we mean that A B is positive, while A > B means that A B is strictly positive. An operator C is an isometry if C C = I, a contraction if C C I and an expansive operator if C C I. A linear map Φ on BH is positive if ΦA for each A and is strictly positive if ΦA > for each A >. A positive linear map Φ is strictly positive if and only if ΦI > [8]. A continuous function f defined on an interval J is said to be operator convex if fλa+1 λb λfa+1 λfb for all λ [,1] and all self-adjoint operators A,B with spectra in J. Hansen and Pedersen see [6, Theorem 1.9] presented an operator version of the Jensen inequality for an operator convex function by establishing that if f is operator convex on J, then 1.3 fc AC C fac for any self-adjoint operator A with spectrum in J and any isometry C. Several versions of the Jensen operator inequality can be found in [6, 8]. Among them, we are interested in the following generalization of 1.3: n f Φ i A i Φ i fa i, in which the operators A i i = 1,...,n are self-adjoint with spectra in J and Φ 1,...,Φ n are positive linear maps on BH with n Φ ii = I [14]. Recently, J. Mićić, Z. Pavić and J. Pečarić[13] obtained a Jensen inequality for operators without the assumption of operator convexity. Regarding the possible operator extensions of 1., there are some interesting works. Letf beacontinuousconvexfunctionon[m,m],a 1,...,A n beself-adjointoperators with spectrain [m,m] and Φ 1,...,Φ n be positive linear maps with n Φ ii = I. The following operator version of the Mercer inequality 1. was proved in [1]: 1.4 f M +m Φ i A i fm+fm Φ i fa i.

4 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October M. Kian and M.S. Moslehian Furthermore, some refinements, applications and reformulations of inequality 1.4 for some other types of functions have been obtained in [3, 7, 9, 11]. The function f : [m,m] [m,m ] R is separately operator convex if the functionsg t andh s definedbyg t s = ft,s = h s tareoperatorconvex,respectively, on [m,m ] and [m,m] for each t [m,m] and s [m,m ]. In Section, we present a Hermite Hadamard Mercer type inequality and then generalize it for Hilbert space operators. In Section 3, we obtain another variant of inequality 1.4. Also we give a Jensen Mercer operator type inequality for separately operator convex functions.. Hermite Hadamard Mercer type inequalities. In this section, we present a Hermite Hadamard type inequality using the Mercer inequality 1. and then give its operator extension..1 and. Theorem.1. Let f be a convex function on [m,m]. Then f M +m x+y fm+fm ftx+1 tydt x+y fm+fm f, f M +m x+y for all x,y [m,m]. 1 y x y x fm +m tdt fm+fm fx+fy Proof. It follows from the Jensen Mercer inequality that f M +m a+b.3 fm+fm fa+fb for each a,b [m,m]. Let t [,1] and x,y [m,m]. Replacing a and b respectively by tx+1 ty and 1 tx+ty in.3, we obtain f M +m x+y.4 fm+fm ftx+1 ty+f1 tx+ty. By integrating both sides of.4, we get f M +m x+y fm+fm.5 1 ftx+1 ty+f1 tx+tydt.

5 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October 13 Operator Jensen Mercer inequality 745 Due to.6 ftx+1 tydt = f1 tx+tydt = 1 y x y x ftdt, inequality.5 gives rise to the first inequality of.1. The second inequality of.1 follows directly from the Hermite Hadamard inequality. Next we prove inequality.. The Hermite Hadamard inequality implies that fm +m tx+1 tydt = ftm +m x+1 tm +m ydt M +m x+m +m y f = f M +m x+y.7. On the other hand, the Mercer inequality gives.8 fm +m tx+1 ty fm+fm tfx+1 tfy. Integrating both sides of.8 we get.9 fm +m tx+1 tydt fm+fm = fm+fm fx+fy. tfx+1 tfydt Inequality. now follows immediately from inequalities.6,.7 and The following operator version of inequality. holds true. Theorem.. If f is convex on [m,m], then fm +m tφa+1 tφbdt fm+fm ΦfA+ΦfB, f m+m ΦA+ΦB fm+fm ftφa+1 tφbdt for all self-adjoint operators A,B with spectra in [m,m] and a unital positive linear map Φ. Furthermore, if f is operator convex, then f M +m ΦA+ΦB fm +m tφa+1 tφbdt.1 fm+fm ΦfA+ΦfB.

6 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October M. Kian and M.S. Moslehian Proof. First note that since f is continuous, the vector valued integrals such as.1 exist for all self-adjoint operators A and B with spectra in [m,m]. We have fm +m tφa+1 tφbdt [fm+fm tφfa 1 tφfb]dt by the Jensen Mercer operator inequality 1.4 = fm+fm ΦfA+ΦfB, which is the desired inequality.1. Moreover, using the Jensen Mercer operator inequality, we get f m+m ΦA+ΦB = f m+m tφa+1 tφb+1 tφa+tφb fm+fm ftφa+1 tφb+f1 tφa+tφb. Integrating from both sides of the later inequality leads us to.11. If f is operator convex, then f M +m ΦA+ΦB M +m tφa 1 tφb+m +m tφb 1 tφa = f.13 1 [fm +m tφa 1 tφb+fm +m tφb 1 tφa]. Integrating from both sides of inequality.13 we get the first inequality of.1. The second one is clear. Example.3. If f : J R is a convex function, then the inequality A+B 1 f fta+1 tbdt fa+fb may not hold in general [15]. To see this, consider the convex function ft = t 4 which appears in some counter-examples, starting with a work of M.-D. Choi [4], and Hermitian matrices A = 1 1 1, B = Nevertheless, inequalities.1 and.11 are valid for all m, M J provided that the spectra of A and B are contained in [m,m]..

7 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October 13 Operator Jensen Mercer inequality A variant of the Jensen Mercer inequality for operators. We use an idea from [17] to obtain one of our main results. Theorem 3.1. Let A i i = 1,...,n be positive operators acting on a finite dimensional Hilbert space H with n A i = I. If f is convex on an interval [m,m] containing, then 3.1 f M +m x i A i fm+fm fx i A i for all x 1,...,x n [m,m]. Proof. Clearly, the spectrum of M+m n x ia i is contained in [m,m]. Without loss of generality we may assume that f = if f we may consider the convex function gx = fx f instead of f. There is a Hilbert space H containing H and a family of mutually orthogonal projections P i i = 1,...,n on H such that n P i = I H and A i = PP i P H for each i = 1,...,n, where P is the projection from H onto H [17]. Therefore, f M +m x i A i = f M +m x i PP i P H n = f M +m P x i P i P H I PI P H n fm+fm Pf x i P i P H I PfI P H by Jensen Mercer operator inequality 1.4 with Φ 1 A = PAP and Φ A = 1 PA1 P n fm+fm P fx i P i P H by f = and the spectral theorem = fm+fm fx i A i. Corollary 3.. Let f and x i i = 1,...,n be as in Theorem 3.1 and f =. If n A i I, then inequality 3.1 remains true. Proof. Put B = I n A i. Then B + n A i = I. Hence, f M +m x i A i = f M +m x i A i B

8 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October M. Kian and M.S. Moslehian fm+fm fm+fm fx i A i fb by 3.1 fx i A i by f =. The following particular case of 3.1 is of special interest. Corollary 3.3. Let A i i = 1,...,n be positive operators acting on a finite dimensional Hilbert space H with n A i = I. If f is convex on the interval [m,m] containing 1 and 1, then f M +m+ A i A i fm+fm f 1 A i f1 A i. i=k+1 i=k+1 Remark 3.4. It should be mentioned that 3.1 implies a weaker version of the Jensen Mercer operator inequality. Assume that X i i = 1,...,n are selfadjoint operators on a finite dimensional Hilbert space H with spectra in [m,m] and w i [,1] with n w i = 1. Let X i = k λ ijp ij i = 1,...,n be the spectral decomposition of X i so that λ ij [m,m]. It follows from n k w ip ij = I that f M +m w i X i = f M +m w i λ ij P ij fm+fm = fm+fm = fm+fm fλ ij w i P ij by 3.1 w i f λ ij P ij w i fx i. Let f be an operator convex function with f. It follows from the Jensen operator inequality 1.3 that 3. C fac fc AC for any invertible expansive operator C and any self-adjoint operator A. Theorem 3.5. Let m < M and let Φ be a positive linear map on BH with < ΦI I. Let m = min{m ΦIx,x ; x = 1} and M = max{m ΦIx,x ; x = 1}.

9 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October 13 Operator Jensen Mercer inequality 749 If f : J R is an operator convex function with f and [m,m] [m,m ] J, then fm+mφi ΦA fm+fm ΦfA for any self-adjoint operator A with spectrum contained in [m, M]. Proof. Define the positive linear map Ψ on BH by ΨX = ΦI 1 ΦXΦI 1, X BH. Then Ψ is unital, and it follows from 1.4 that fm+m ΨA fm+fm ΨfA for each self-adjoint operator A with spectrum in [m, M]. Therefore, Hence, fm+m ΦI 1 ΦAΦI 1 fm+fm ΦI 1 ΦfAΦI fφi 1 m+mφi ΦAΦI 1 ΦI 1 fm+fmφi ΦfAΦI 1. On the other hand, ΦI 1 is an expansive operator. Using 3. we obtain 3.4 ΦI 1 fm+mφi ΦAΦI 1 fφi 1 m+mφi ΦAΦI 1. Now, the result follows from inequalities 3.3 and 3.4. Corollary 3.6. Let f and H be as in Theorem 3.5. Let Φ i i = 1,...,n be positive linear maps on BH with < ΦI = n Φ ii I. Then f m+mφi Φ i A i fm+fm Φ i fa i for all self-adjoint operators A i with spectra in [m,m]. Proof. Assume that A 1,...,A n are self-adjoint operators on H with spectra in [m,m] and Φ 1,...,Φ n are positive linear maps on BH with < n Φ ii I. For A,B BH assume that A B is the operator defined on BH H by A. Now apply Theorem 3.5 to the self-adjoint operator A on the Hilbert B space H H defined by A = A 1 A n and the positive linear map Φ defined on BH H by ΦA = Φ 1 A 1 Φ n A n.

10 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October M. Kian and M.S. Moslehian The next theorem yields a Jensen Mercer operator type inequality for separately operator convex functions. Theorem 3.7. Let f : [m,m] [m,m ] R be a separately operator convex function. Let Φ i,ψ j, 1 i r,1 j k be positive linear maps on BH with r Φ ii = I = k Ψ ji. Then f M +m Φ i A i,m +m Ψ j B j fm,m +fm,m +fm,m +fm,m f Φ i A i, M +m f m+m, Ψ j B j +f Φ i A i, Ψ j B j for all self-adjoint operators A i with spectra in [m,m] and B j with spectra in [m,m ]. Proof. Since f is separately convex, we have 3.5 fm+m t,s fm,s+fm,s ft,s, and 3.6 ft,m +m s ft,m +ft,m ft,s for all t [m,m] and s [m,m ]. Adding inequalities 3.5 and 3.6 we obtain ft,s 1 [fm,s+fm,s+ft,m +ft,m 3.7 ft,m +m s fm+m t,s], for all t [m,m] and s [m,m ]. Using functional calculus for inequality 3.7 we get f M +m Φ ia i,m +m Ψ jb j f f [ f m,m +m M +m M +m Ψ jb j +f M,M +m Φ ia i,m +f M +m Ψ jb j Φ ia i,m Φ ia i, Ψ jb j f Φ ia i,m +m ] Ψ jb j.

11 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October 13 Operator Jensen Mercer inequality 751 Since f is separatelyconvex, the functions g s and h t defined byg s t = ft,s = h t s are convex on [m,m] and [m,m ] respectively. It follows from 1.4 that 3.9 f m,m +m Ψ j B j fm,m +fm,m Ψ j fm,b j, 3.1 f M,M +m Ψ j B j fm,m +fm,m Ψ j fm,b j, f f M +m M +m Φ i A i,m fm,m +fm,m Φ i A i,m fm,m +fm,m Summing inequalities 3.9, 3.1, 3.11 and 3.1 we obtain f m,m +m Ψ j B j +f M,M +m Ψ j B j +f M +m Φ i A i,m +f M +m Φ i fa i,m, Φ i fa i,m. Φ i A i,m [fm,m +fm,m +fm,m +fm,m ] Ψ j fm,b j +fm,b j Φ i fa i,m +fa i,m [fm,m +fm,m +fm,m +fm,m ] by the convexity m+m Ψ j f,b j + Φ i f A i, M +m [fm,m +fm,m +fm,m +fm,m ] f m+m 3.13, Ψ j B j +f by the operator convexity. Φ i A i, M +m Also, since f is separately operator convex, we have 1 f M +m Φ i A i, Ψ j B j + 1 f Φ i A i, Ψ j B j

12 Electronic Journal of Linear Algebra ISSN Volume 6, pp , October M. Kian and M.S. Moslehian 3.14 and f f Φ i A i,m +m f M +m, Ψ j B j, Ψ j B j + 1 f Φ i A i, Φ i A i, M +m. Ψ j B j Combining inequalities 3.8, 3.13, 3.14 and 3.15, one can easily conclude the desired result. Acknowledgment. The authors would like to thank the referees for several useful comments improving the paper. REFERENCES [1] S. Abramovich, J. Barić, and J. Pečarić. A variant of Jessen s inequality of Mercer s type for superquadratic functions. J. Inequal. Pure Appl. Math., 93:13, Article 6, 8. [] J. Barić and A. Matković. Bounds for the normalized Jensen Mercer functional. J. Math. Inequal., 34:59 541, 9. [3] J. Barić, A. Matković, and J. Pečarić. A variant of the Jensen Mercer operator inequality for superquadratic functions. Math. Comput. Modelling, 51:13 139, 1. [4] M.-D. Choi. A Schwarz inequality for positive linear maps on C -algebras. Illinois J. Math., Volume 18: , [5] S.S. Dragomir. Hermite-Hadamards type inequalities for operator convex functions. Appl. Math. Comput., 183:766 77, 11. [6] T. Furuta, H. Mićić, J. Pečarić, and Y. Seo. Mond Pečarić Method in Operator Inequalities. Zagreb Element, 5. [7] S. Ivelić, A. Matković, and J.E. Pečarić. On a Jensen Mercer operator inequality. Banach J. Math. Anal., 51:19 8, 11. [8] M. Khosravi, J.S. Aujla, S.S. Dragomir, and M.S. Moslehian. Refinements of Choi Davis Jensen s inequality. Bull. Math. Anal. Appl., 3:17 133, 11. [9] A. Matković and J. Pečarić. On a variant of the Jensen Mercer inequality for operators. J. Math. Inequal., 3:99 37, 8. [1] A. Matković, J. Pečarić, and I. Perić. A variant of Jensen s inequality of Mercer s type for operators with applications. Linear Algebra Appl., 418: , 6. [11] A. Matković, J. Pečarić, and I. Perić. Refinements of Jensen s inequality of Mercer s type for operator convex functions. Math. Inequal. Appl., 111:113 16, 8. [1] A.McD. Mercer. A variant of Jensen s inequality. J. Inequal. Pure Appl. Math., 44:, Article 73, 3. [13] J. Mićić, Z. Pavić, and J. Pečarić. Jensen s inequality for operators without operator convexity. Linear Algebra Appl., 434:18 137, 11. [14] B. Mond and J. Pečarić. Converses of Jensen inequality for several operators. Rev. Anal. Numér. Théor. Approx., 3: , 1994.

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