Cyclic Refinements of the Different Versions of Operator Jensen's Inequality
|
|
- Jordan Francis
- 5 years ago
- Views:
Transcription
1 Electronic Journal of Linear Algebra Volume 31 Volume 31: 2016 Article Cyclic Refinements of the Different Versions of Operator Jensen's Inequality Laszlo Horvath University of Pannonia, Egyetem u. 10, 8200 Veszprem, Hungary, Khuram A. Khan University of Sargodha, Sargodha, Pakistan, Josip Pecaric Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, Zagreb, Croatia, Follow this and additional works at: Recommended Citation Horvath, Laszlo; Khan, Khuram A.; and Pecaric, Josip. 2016, "Cyclic Refinements of the Different Versions of Operator Jensen's Inequality", Electronic Journal of Linear Algebra, Volume 31, pp 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
2 CYCLIC REFINEMENTS OF THE DIFFERENT VERSIONS OF OPERATOR JENSEN INEQUALITY LÁSZLÓ HORVÁTH, KHURAM ALI KHAN, AND JOSIP PEČARIĆ Abstract. Refinements of the operator Jensen inequality for convex and operator convex functions are given by using cyclic refinements of the discrete Jensen inequality. Similar refinements are fairly rare in the literature. Some applications of the results to norm inequalities, to the Hölder- McCarthy inequality and to generalized weighted power means for operators are presented. Key words. Operator Jensen inequality, Operator monotone, Operator mean. AMS subject classifications. 47A63, 26A Introduction. In this paper, H,, denotes a complex Hilbert space. The C -algebra of all bounded linear operators on H will be denoted by BH. We always understand the norm of an operator A BH as A := sup Ax. x 1 The identity operatoron H is denoted by I H. The spectrum of anoperatora BH is denoted by spa. An operator A BH is called positive, if Ax,x 0 for every x H, or equivalently, A is self-adjoint and spa [0, [. An operator A BH is called strictly positive, if it is positive and invertible. For an interval J R, SJ means the class of all self-adjoint operators from BH, whose spectra are contained in J. Let J R be an interval, and f : J R be a function. The function f is called convex on J if f λx+1 λy λfx+1 λfy for all x, y J and for all 0 λ 1. If f is continuous on J, and A SJ, then f A is defined by the continuous functional calculus for self-adjoint operators see Received by the editors on August 27, Accepted for publication on February 23, Handling Editor: Harm Bart. Department of Mathematics, University of Pannonia, Egyetem u. 10, 8200 Veszprém, Hungary lhorvath@almos.uni-pannon.hu. Department of Mathematics, University of Sargodha, Sargodha, Pakistan khuramsms@gmail.com. Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, Zagreb, Croatia pecaric@mahazu.hazu.hr. 125
3 126 L. Horváth, K.A. Khan, and J. Pečarić Rudin [14]. The function f is said to be operator monotone increasing on J if f is continuous on J and A, B SJ, A B i.e. B A is a positive operator implies fa fb. The function f is called operator convex on J if f is continuous on J and for all A, B SJ and for all λ [0,1]. fλa+1 λb λfa+1 λfb We say that the numbers p 1,...,p n represent a positive discrete probability distribution if p i > 0 p i 0 1 i n and n p i = 1. The following well known results are operator versions of Jensen inequality: Theorem 1.1. Operator Jensens inequality for convex functions, [3, 11] Let J R be an interval. Let A i SJ and x i H i = 1,...,n with n x i 2 = 1. If f : J R is continuous and convex, then 1.1 f A i x i,x i n fa i x i,x i. Theorem 1.2. Operator Jensens inequality for operator convex functions, [12] Let J R be an interval, and K be a complex Hilbert space. Let A i SJ i = 1,...,n, Φ i : BH BK i = 1,...,n be unital positive linear maps, and let p 1,...,p n represent a discrete probability distribution. If f : J R is operator convex, then n 1.2 f p i Φ i A i p i Φ i f A i. A linear map Φ : BH BK is positive if ΦA is positive for all positive A BH, and unital if ΦI H = ΦI K. Φ is called strictly positive if ΦA is strictly positive for all strictly positive A BH. It is given new cyclic refinements of the discrete Jensen inequality in the papers Brnetić, Khan and Pečarić[2] and Horváth, Khan and Pečarić[5]. Moreover, we refer [1] for numerical inequalities. In this paper, we obtain refinements of 1.1 and 1.2 in the spirit of [5]. Refinements of operator versions of Jensen inequality has been less extensively studied than refinements of the discrete or the integral form of Jensen inequality. For some results, we refer to the papers Khosravi, Aujla, Dragomir and Moslehian [9], Niezgoda [13], Khan and Hanif [7], Kian and Moslehian [8], and the
4 Cyclic Refinements of the Different Versions of Operator Jensen Inequality 127 book Horváth, Khan and Pečarić [4]. Some applications are also given: Refinements of norm inequalities and the Hölder-McCarthy inequality; introduction of some mixed symmetric means for operators and investigation of their monotonicity properties. 2. Refinements of the operator Jensen inequality for convex functions. In the sequel, we shall use the following convention: Let 2 k n be integers, i {1,...,n} and j {0,...,k 1}; if i+j > n, then i+j means i+j n. Our first result a new refinement of the operator Jensen inequality for convex functions: Theorem 2.1. Let 2 k n be integers, let x := x 1,...,x n H n such that x i 0 i = 1,...,n and n x i 2 = 1, and let λ := λ 1,...,λ k represent a positive discrete probability distribution. Let J R be an interval, A i SJ i = 1,...,n and A := A 1,...,A n. If f : J R is continuous and convex, then = f A i x i,x i D c = D c f,a,x,λ n λ j+1 x i+j 2 f n f A i x i,x i. 1 λ j+1 A i+j x i+j,x i+j λ j+1 x i+j 2 Proof. Since λj+1 x i+j 1/2 λ j+1 x i+j 2 2 = 1, the operator Jensen inequality for convex functions yields n D c = λ j+1 x i+j 2
5 128 L. Horváth, K.A. Khan, and J. Pečarić = f A i+j λj+1 x i+j, 1/2 λ j+1 x i+j 2 λj+1 x i+j 1/2 λ j+1 x i+j 2 n n k λ j+1 f A i+j x i+j,x i+j = f A i x i,x i n f A i x i,x i. Conversely, it is easy to check that n λ j+1 x i+j 2 = 1, and therefore, the convexity of f implies n k D c f λ j+1 A i+j x i+j,x i+j = f A i x i,x i = f A i x i,x i. j=1 j=1 λ j λ j The following particular case is interesting. Corollary 2.2. Let 2 k n be integers, let x H with x = 1, and let λ 1,...,λ k and p 1,...,p n represent positive discrete probability distributions. Let J R be an interval, and A i SJ i = 1,...,n. If f : J R is continuous and convex, then: a n f p i A i x,x n f n p i f A i x,x. 1 A i+j x,x
6 Cyclic Refinements of the Different Versions of Operator Jensen Inequality 129 b In case of A := A 1 = = A n, f Ax,x n f f Ax,x. 1 Ax,x Proof. a Theorem 2.1 can be applied to the vectors x i := p i x i = 1,...,n. b It is a special case of a. Some norm inequalities can be obtained from Corollary 2.2 a. Corollary 2.3. Let 2 k n be integers, and let λ 1,...,λ k and p 1,...,p n represent positive discrete probability distributions. Let J [0, [ be an interval, and A i SJ i = 1,...,n. If f : J R is nonnegative, continuous, increasing and convex, then n n f p i A i 1 f A i+j λ j+1 p i+j n p i f A i. Proof. If A BH is a positive operator, then A = sup Ax,x. By using x =1 this, the continuity and the increase of f, and Corollary 2.2 a, we have n f p i A i = f sup sup x =1 x =1 sup x =1 n n p i A i x,x = sup f p i A i x,x n f x =1 1 n n p i f A i x,x = p i f A i. A i+j x,x
7 130 L. Horváth, K.A. Khan, and J. Pečarić Remark 2.4. WeconsidernowsomespecialcasesofCorollary2.3. Let2 k n be integers, and let λ 1,...,λ k and p 1,...,p n represent positive discrete probability distributions. Let J [0, [ be an interval, and A i SJ i = 1,...,n. 2.1 a For α 1, n α p i A i n 1 α A i+j α n p i A α i, and for 0 < α < 1 the reverse inequalities hold. If the operators are strictly positive, 2.1 is also true for α < 0. b By choosing f = exp, we have exp n p i A i n exp n p i expa i. 1 A i+j λ j+1 p i+j From Corollary 2.2, b a refinement of the Hölder-McCarthy inequality see [6] is derived. Corollary 2.5. Let 2 k n be integers, let x H be a unit vector, and let λ 1,...,λ k and p 1,...,p n represent positive discrete probability distributions. Let A BH be a positive operator. Then the following hold: 2.2 a For every α 1, Ax,x α n 1 α α Ax,x A α x,x. b For every 0 < α < 1, Ax,x α n 1 α α Ax,x A α x,x. c If A is strictly positive and α < 0, then 2.2 also holds.
8 Cyclic Refinements of the Different Versions of Operator Jensen Inequality Refinements of the operator Jensen inequality for operator convex functions. In the next result, we obtain a new refinement for operator Jensen inequality for operator convex functions. Theorem 3.1. Let 2 k n be integers, and let λ := λ 1,...,λ k and p := p 1,...,p n represent positive discrete probability distributions. Let J R be an interval, A i SJ i = 1,...,n and A := A 1,...,A n. Let K be a complex Hilbert space, Φ i : BH BK i = 1,...,n be unital positive linear maps, and Φ := Φ 1,...,Φ n. If f : J R is operator convex, then f p i Φ i A i D oc = D oc f,a,φ,p,λ := n f n p i Φ i f A i. Φ i+j A i+j Proof. The operator Jensen inequality for operator convex functions shows that n D oc Φ i+j f A i+j n k = p i Φ i f A i = λ j j=1 n p i Φ i f A i. Since n = 1, we can apply the operator Jensen inequality for operator convex functions again, and have n n D oc f Φ i+j A i+j = f p i Φ i A i. In the following variant of the previous result, the maps Φ 1,...,Φ n are defined directly in terms of unitary operators.
9 132 L. Horváth, K.A. Khan, and J. Pečarić Corollary 3.2. Let 2 k n be integers, and let λ := λ 1,...,λ k and p := p 1,...,p n represent positive discrete probability distributions. Let J R be an interval, A i SJ i = 1,...,n and A := A 1,...,A n. Let C i BH i = 1,...,n be unitary operators. If f : J R is operator convex, then f p i CiA i C i n f n p i Ci f A ic i. Ci+j A i+jc i+j Proof. For every i = 1,...,n, the map Φ i : BH BH defined by Φ i A = C i AC i is a unital positive linear map, and hence Theorem 3.1 can be applied. As an application, we present some monotonicity results for operator means. Let A i BH i = 1,...,n be strictly positive operators, A := A 1,...,A n, and let p := p 1,...,p n represent a positive discrete probability distribution. Let K be a complex Hilbert space, Φ i : BH BK i = 1,...,n be unital strictly positive linear maps, and Φ := Φ 1,...,Φ n. The generalized weighted power mean of the operators A i i = 1,...,n is defined by see [10] M [α] n A,Φ,p = M n [α] A 1,...,A n ;Φ 1,...,Φ n ;p 1,...,p n 1/α := p i Φ i A α i, α R\{0}. Theorem 3.3. Let 2 k n be integers, and let λ := λ 1,...,λ k and p := p 1,...,p n represent positive discrete probability distributions. Let A i BH i = 1,...,n be strictly positive operators, A := A 1,...,A n. Let K be a complex Hilbert space, Φ i : BH BK i = 1,...,n be unital strictly positive linear maps, and Φ := Φ 1,...,Φ n. Then n 1/α p i Φ i A α i M [α,β] 3.1 n = M n [α,β] A,Φ,p,λ n := β/α Φ i+j A α i+j λ j+1 p i+j 1/β
10 Cyclic Refinements of the Different Versions of Operator Jensen Inequality 133 n p i Φ i A 1/β β i, if either α β 1 or 1 β α or 1 β and α β 2α. The reverse inequalities hold in 3.1 if either 1 β α or α β 1 or β 1 and 2α β α. Proof. The following properties of the function g : ]0, [ R, gx = x r are well known see [3]: It is operator convex if either 1 r 2 or 1 r 0, and g is operator convex if 0 r 1; g is operator monotone increasing if 0 r 1 and operator monotone decreasing if 1 r 0. Byusingtheseproperties,Theorem3.1canbeappliedtothefunctionf : ]0, [ R, f x = x β/α and the operators A α i i = 1,...,n. Remark 3.4. M n [α,β] can be considered as the mixed symmetric mean corresponding to D oc in Theorem 3.1. REFERENCES [1] S. Abramovich. Convexity, subadditivity and generalized Jensen s inequality. Ann. Funct. Anal., 4: , [2] I. Brnetić, K.A. Khan, and J. Pečarić. Refinement of Jensen s inequality with applications to cyclic mixed symmetric means and Cauchy means. J. Math. Inequal., 9: , [3] T. Furuta, J. Mićić, J. Pečarić, and Y. Seo. Mond-Pečarić Method in Operator Inequalities, Inequalities for Bounded Selfadjoint Operators on a Hilbert Space. Element, Zagreb, [4] L. Horváth, K.A. Khan, and J. Pečarić. Combinatorial Improvements of Jensen s Inequality, Classical and New Refinements of Jensen s Inequality with Applications. Element, Zagreb, [5] L. Horváth, K.A. Khan, and J. Pečarić. Cyclic refinements of the discrete and integral form of Jensen s inequality with applications. Analysis, to appear. [6] C.A. McCarthy. c p. Israel J. Math., 5: , [7] M.K. Khan and M. Hanif. On some refinements of Jensen s inequality. J. Approx. Theory, 93: , [8] M. Kian and M.S. Moslehian. Refinements of operator Jensen-Mercer inequality. Electron. J. Linear Algebra, 26: , [9] M. Khosravi, J.S. Aujla, S.S. Dragomir, and M.S. Moslehian. Refinements of Choi-Davis-Jensen s inequality. Bull. Math. Anal. Appl., 3: , [10] J. Mićić and J. Pečarić. Order among power means of positive operators. Sci. Math. Jpn., , e [11] B. Mond and J. Pečarić. Convex inequalities in Hilbert space. Houston J. Math., 19: , [12] B. Mond and J. Pečarić. Converses of Jensen s inequality for several operators. Rev. Anal. Numér. Théor. Approx., 23: , [13] M. Niezgoda. Choi-Davis-Jensen s inequality and generalized inverses of linear operators. Electron. J. Linear Algebra, 26: , [14] W. Rudin. Functional Analysis. McGraw-Hill, Inc., 1991.
Refinements of the operator Jensen-Mercer inequality
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
More informationThis is a submission to one of journals of TMRG: BJMA/AFA EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY WITH CONDITION ON SPECTRA
This is a submission to one of journals of TMRG: BJMA/AFA EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY WITH CONDITION ON SPECTRA JADRANKA MIĆIĆ, JOSIP PEČARIĆ AND JURICA PERIĆ3 Abstract. We give
More informationOn (T,f )-connections of matrices and generalized inverses of linear operators
Electronic Journal of Linear Algebra Volume 30 Volume 30 (2015) Article 33 2015 On (T,f )-connections of matrices and generalized inverses of linear operators Marek Niezgoda University of Life Sciences
More informationJENSEN S OPERATOR INEQUALITY AND ITS CONVERSES
MAH. SCAND. 100 (007, 61 73 JENSEN S OPERAOR INEQUALIY AND IS CONVERSES FRANK HANSEN, JOSIP PEČARIĆ and IVAN PERIĆ (Dedicated to the memory of Gert K. Pedersen Abstract We give a general formulation of
More informationarxiv: v1 [math.fa] 30 Oct 2011
AROUND OPERATOR MONOTONE FUNCTIONS MOHAMMAD SAL MOSLEHIAN AND HAMED NAJAFI arxiv:111.6594v1 [math.fa] 3 Oct 11 Abstract. We show that the symmetrized product AB + BA of two positive operators A and B is
More informationELA CHOI-DAVIS-JENSEN S INEQUALITY AND GENERALIZED INVERSES OF LINEAR OPERATORS
CHOI-DAVIS-JENSEN S INEQUALITY AND GENERALIZED INVERSES OF LINEAR OPERATORS MAREK NIEZGODA Abstract. In this paper, some extensions of recent results on Choi-Davis-Jensen s inequality due to Khosravi et
More informationJensen s Operator and Applications to Mean Inequalities for Operators in Hilbert Space
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. 351 01 1 14 Jensen s Operator and Applications to Mean Inequalities for Operators in Hilbert
More informationJENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES FOR OPERATORS IN HILBERT SPACE
JENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES FOR OPERATORS IN HILBERT SPACE MARIO KRNIĆ NEDA LOVRIČEVIĆ AND JOSIP PEČARIĆ Abstract. In this paper we consider Jensen s operator which includes
More informationKantorovich type inequalities for ordered linear spaces
Electronic Journal of Linear Algebra Volume 20 Volume 20 (2010) Article 8 2010 Kantorovich type inequalities for ordered linear spaces Marek Niezgoda bniezgoda@wp.pl Follow this and additional works at:
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics KANTOROVICH TYPE INEQUALITIES FOR 1 > p > 0 MARIKO GIGA Department of Mathematics Nippon Medical School 2-297-2 Kosugi Nakahara-ku Kawasaki 211-0063
More informationAnn. Funct. Anal. 5 (2014), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:www.emis.
Ann. Funct. Anal. 5 (2014), no. 2, 147 157 A nnals of F unctional A nalysis ISSN: 2008-8752 (electronic) URL:www.emis.de/journals/AFA/ ON f-connections OF POSITIVE DEFINITE MATRICES MAREK NIEZGODA This
More informationOPERATOR INEQUALITIES RELATED TO WEAK 2 POSITIVITY
Journal of Mathematical Inequalities Volume 7, Number 2 2013, 175 182 doi:10.7153/jmi-07-17 OPERATOR INEQUALITIES RELATED TO WEAK 2 POSITIVITY MOHAMMAD SAL MOSLEHIAN AND JUN ICHI FUJII Communicated by
More informationIMPROVED ARITHMETIC-GEOMETRIC AND HEINZ MEANS INEQUALITIES FOR HILBERT SPACE OPERATORS
IMPROVED ARITHMETI-GEOMETRI AND HEINZ MEANS INEQUALITIES FOR HILBERT SPAE OPERATORS FUAD KITTANEH, MARIO KRNIĆ, NEDA LOVRIČEVIĆ, AND JOSIP PEČARIĆ Abstract. The main objective of this paper is an improvement
More informationInequalities among quasi-arithmetic means for continuous field of operators
Filomat 26:5 202, 977 99 DOI 0.2298/FIL205977M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Inequalities among quasi-arithmetic
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 4 (200), no., 59 69 Banach Journal of Mathematical Analysis ISSN: 735-8787 (electronic) www.emis.de/journals/bjma/ IMPROVEMENT OF JENSEN STEFFENSEN S INEQUALITY FOR SUPERQUADRATIC
More informationResearch Article Refinements of the Lower Bounds of the Jensen Functional
Abstract and Applied Analysis Volume 20, Article ID 92439, 3 pages doi:0.55/20/92439 Research Article Refinements of the Lower Bounds of the Jensen Functional Iva Franjić, Sadia Khalid, 2 and Josip Pečarić
More informationInequalities involving eigenvalues for difference of operator means
Electronic Journal of Linear Algebra Volume 7 Article 5 014 Inequalities involving eigenvalues for difference of operator means Mandeep Singh msrawla@yahoo.com Follow this and additional works at: http://repository.uwyo.edu/ela
More informationarxiv: v1 [math.fa] 13 Oct 2016
ESTIMATES OF OPERATOR CONVEX AND OPERATOR MONOTONE FUNCTIONS ON BOUNDED INTERVALS arxiv:1610.04165v1 [math.fa] 13 Oct 016 MASATOSHI FUJII 1, MOHAMMAD SAL MOSLEHIAN, HAMED NAJAFI AND RITSUO NAKAMOTO 3 Abstract.
More informationBuzano Inequality in Inner Product C -modules via the Operator Geometric Mean
Filomat 9:8 (05), 689 694 DOI 0.98/FIL508689F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Buzano Inequality in Inner Product
More informationJ. Mićić, J. Pečarić () Operator means for convex functions Convexity and Applications 1 / 32
J. Mićić, J. Pečarić () Operator means for convex functions Convexity and Applications 1 / 32 Contents Contents 1 Introduction Generalization of Jensen s operator inequality Generalization of converses
More informationHERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR GEOMETRICALLY CONVEX FUNCTIONS
HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR GEOMETRICALLY CONVEX FUNCTIONS A TAGHAVI, V DARVISH, H M NAZARI, S S DRAGOMIR Abstract In this paper, we introduce the concept of operator geometrically
More informationTHE HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR CONVEX FUNCTIONS
THE HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR CONVEX FUNCTIONS S.S. DRAGOMIR Abstract. Some Hermite-Hadamard s type inequalities or operator convex unctions o seladjoint operators in Hilbert spaces
More informationON SOME GENERALIZED NORM TRIANGLE INEQUALITIES. 1. Introduction In [3] Dragomir gave the following bounds for the norm of n. x j.
Rad HAZU Volume 515 (2013), 43-52 ON SOME GENERALIZED NORM TRIANGLE INEQUALITIES JOSIP PEČARIĆ AND RAJNA RAJIĆ Abstract. In this paper we characterize equality attainedness in some recently obtained generalized
More informationSOME HERMITE HADAMARD TYPE INEQUALITIES FOR THE PRODUCT OF TWO OPERATOR PREINVEX FUNCTIONS
Banach J. Math. Anal. 9 15), no., uncorrected galleyproo DOI: 1.1535/bjma/9-- ISSN: 1735-8787 electronic) www.emis.de/journals/bjma/ SOME HERMITE HADAMARD TYPE INEQUALITIES FOR THE PRODUCT OF TWO OPERATOR
More informationA Diaz-Metcalf type inequality for positive linear maps and its applications
Electronic Journal of Linear Algebra Volume Volume 0) Article 0 A Diaz-Metcalf type inequality for positive linear maps and its applications Mohammad Sal Moslehian Ritsuo Nakamoto Yuki Seo Follow this
More informationTHE HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR GODUNOVA-LEVIN CLASS OF FUNCTIONS IN HILBERT SPACE
Ordu Üniv. Bil. Tek. Derg., Cilt:6, Sayı:, 06,50-60/Ordu Univ. J. Sci. Tech., Vol:6, No:,06,50-60 THE HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR GODUNOVA-LEVIN CLASS OF FUNCTIONS IN HILBERT SPACE
More informationA norm inequality for pairs of commuting positive semidefinite matrices
Electronic Journal of Linear Algebra Volume 30 Volume 30 (205) Article 5 205 A norm inequality for pairs of commuting positive semidefinite matrices Koenraad MR Audenaert Royal Holloway University of London,
More informationResearch Article Extension of Jensen s Inequality for Operators without Operator Convexity
Abstract and Applied Analysis Volume 2011, Article ID 358981, 14 pages doi:10.1155/2011/358981 Research Article Extension of Jensen s Inequality for Operators without Operator Convexity Jadranka Mićić,
More informationGENERALIZATIONS AND IMPROVEMENTS OF AN INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA. Sadia Khalid and Josip Pečarić
RAD HAZU. MATEMATIČKE ZNANOSTI Vol. 18 = 519 014: 73-89 GENERALIZATIONS AND IMPROVEMENTS OF AN INEQUALITY OF HARDY-LITTLEWOOD-PÓLYA Sadia Khalid and Josip Pečarić Abstract. Some generalizations of an inequality
More informationImprovements of the Giaccardi and the Petrović inequality and related Stolarsky type means
Annals of the University of Craiova, Mathematics and Computer Science Series Volume 391, 01, Pages 65 75 ISSN: 13-6934 Improvements of the Giaccardi and the Petrović inequality and related Stolarsky type
More informationSingular Value and Norm Inequalities Associated with 2 x 2 Positive Semidefinite Block Matrices
Electronic Journal of Linear Algebra Volume 32 Volume 32 (2017) Article 8 2017 Singular Value Norm Inequalities Associated with 2 x 2 Positive Semidefinite Block Matrices Aliaa Burqan Zarqa University,
More informationOperator inequalities associated with A log A via Specht ratio
Linear Algebra and its Applications 375 (2003) 25 273 www.elsevier.com/locate/laa Operator inequalities associated with A log A via Specht ratio Takayuki Furuta Department of Mathematical Information Science,
More informationITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N ( ) 528
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 40 2018 528 534 528 SOME OPERATOR α-geometric MEAN INEQUALITIES Jianing Xue Oxbridge College Kuning University of Science and Technology Kuning, Yunnan
More informationarxiv: v1 [math.fa] 1 Sep 2014
SOME GENERALIZED NUMERICAL RADIUS INEQUALITIES FOR HILBERT SPACE OPERATORS MOSTAFA SATTARI 1, MOHAMMAD SAL MOSLEHIAN 1 AND TAKEAKI YAMAZAKI arxiv:1409.031v1 [math.fa] 1 Sep 014 Abstract. We generalize
More informationPerturbations of functions of diagonalizable matrices
Electronic Journal of Linear Algebra Volume 20 Volume 20 (2010) Article 22 2010 Perturbations of functions of diagonalizable matrices Michael I. Gil gilmi@bezeqint.net Follow this and additional works
More informationMoore-Penrose Inverse and Operator Inequalities
E extracta mathematicae Vol. 30, Núm. 1, 9 39 (015) Moore-Penrose Inverse and Operator Inequalities Ameur eddik Department of Mathematics, Faculty of cience, Hadj Lakhdar University, Batna, Algeria seddikameur@hotmail.com
More informationSome Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces
Some Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces S.S. Dragomir Abstract. Some new inequalities for commutators that complement and in some instances improve recent results
More informationOn the Feichtinger conjecture
Electronic Journal of Linear Algebra Volume 26 Volume 26 (2013) Article 35 2013 On the Feichtinger conjecture Pasc Gavruta pgavruta@yahoo.com Follow this and additional works at: http://repository.uwyo.edu/ela
More informationMatrix functions that preserve the strong Perron- Frobenius property
Electronic Journal of Linear Algebra Volume 30 Volume 30 (2015) Article 18 2015 Matrix functions that preserve the strong Perron- Frobenius property Pietro Paparella University of Washington, pietrop@uw.edu
More informationON AN INEQUALITY OF V. CSISZÁR AND T.F. MÓRI FOR CONCAVE FUNCTIONS OF TWO VARIABLES
ON AN INEQUALITY OF V. CSISZÁR AND T.F. MÓRI FOR CONCAVE FUNCTIONS OF TWO VARIABLES BOŽIDAR IVANKOVIĆ Faculty of Transport and Traffic Engineering University of Zagreb, Vukelićeva 4 0000 Zagreb, Croatia
More informationOn EP elements, normal elements and partial isometries in rings with involution
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012 Article 39 2012 On EP elements, normal elements and partial isometries in rings with involution Weixing Chen wxchen5888@163.com Follow this
More informationON ZIPF-MANDELBROT ENTROPY AND 3-CONVEX FUNCTIONS
Adv. Oper. Theory https://doi.org/0.535/aot.80-46 ISSN: 538-5X electronic https://projecteuclid.org/aot ON ZIPF-MANDELBROT ENTROPY AND 3-CONVEX FUNCTIONS SADIA KHALID, DILDA PEČARIĆ, and JOSIP PEČARIĆ3
More informationn Inequalities Involving the Hadamard roduct of Matrices 57 Let A be a positive definite n n Hermitian matrix. There exists a matrix U such that A U Λ
The Electronic Journal of Linear Algebra. A publication of the International Linear Algebra Society. Volume 6, pp. 56-61, March 2000. ISSN 1081-3810. http//math.technion.ac.il/iic/ela ELA N INEQUALITIES
More informationarxiv: v1 [math.fa] 19 Aug 2017
EXTENSIONS OF INTERPOLATION BETWEEN THE ARITHMETIC-GEOMETRIC MEAN INEQUALITY FOR MATRICES M. BAKHERAD 1, R. LASHKARIPOUR AND M. HAJMOHAMADI 3 arxiv:1708.0586v1 [math.fa] 19 Aug 017 Abstract. In this paper,
More informationSOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES. S. S. Dragomir
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 5: 011), 151 16 DOI: 10.98/FIL110151D SOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR
More informationarxiv: v1 [math.oa] 21 May 2009
GRAM MATRIX IN C -MODULES LJ. ARAMBAŠIĆ 1, D. BAKIĆ 2 AND M. S. MOSLEHIAN 3 arxiv:0905.3509v1 math.oa 21 May 2009 Abstract. Let (X,, ) be a semi-inner product module over a C -algebra A. For arbitrary
More informationSome Hermite-Hadamard type integral inequalities for operator AG-preinvex functions
Acta Univ. Sapientiae, Mathematica, 8, (16 31 33 DOI: 1.1515/ausm-16-1 Some Hermite-Hadamard type integral inequalities or operator AG-preinvex unctions Ali Taghavi Department o Mathematics, Faculty o
More information2. reverse inequalities via specht ratio To study the Golden-Thompson inequality, Ando-Hiai in [1] developed the following log-majorizationes:
ON REVERSES OF THE GOLDEN-THOMPSON TYPE INEQUALITIES MOHAMMAD BAGHER GHAEMI, VENUS KALEIBARY AND SHIGERU FURUICHI arxiv:1708.05951v1 [math.fa] 20 Aug 2017 Abstract. In this paper we present some reverses
More informationGiaccardi s Inequality for Convex-Concave Antisymmetric Functions and Applications
Southeast Asian Bulletin of Mathematics 01) 36: 863 874 Southeast Asian Bulletin of Mathematics c SEAMS 01 Giaccardi s Inequality for Convex-Concave Antisymmetric Functions and Applications J Pečarić Abdus
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J Math Anal (009), no, 64 76 Banach Journal of Mathematical Analysis ISSN: 75-8787 (electronic) http://wwwmath-analysisorg ON A GEOMETRIC PROPERTY OF POSITIVE DEFINITE MATRICES CONE MASATOSHI ITO,
More informationand monotonicity property of these means is proved. The related mean value theorems of Cauchy type are also given.
Rad HAZU Volume 515 013, 1-10 ON LOGARITHMIC CONVEXITY FOR GIACCARDI S DIFFERENCE J PEČARIĆ AND ATIQ UR REHMAN Abstract In this paper, the Giaccardi s difference is considered in some special cases By
More informationINEQUALITIES FOR THE NORM AND THE NUMERICAL RADIUS OF LINEAR OPERATORS IN HILBERT SPACES
INEQUALITIES FOR THE NORM AND THE NUMERICAL RADIUS OF LINEAR OPERATORS IN HILBERT SPACES S.S. DRAGOMIR Abstract. In this paper various inequalities between the operator norm its numerical radius are provided.
More informationFundamental inequalities and Mond-Pe carić method
Chapter Fundamental inequalities and Mond-Pecarić method In this chapter, we have given a very brief and rapid review of some basic topics in Jensen s inequality for positive linear maps and the Kantorovich
More informationSOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES
SOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES SEVER S DRAGOMIR Abstract Some sharp bounds for the Euclidean operator radius of two bounded linear operators in Hilbert
More informationThe symmetric minimal rank solution of the matrix equation AX=B and the optimal approximation
Electronic Journal of Linear Algebra Volume 18 Volume 18 (2009 Article 23 2009 The symmetric minimal rank solution of the matrix equation AX=B and the optimal approximation Qing-feng Xiao qfxiao@hnu.cn
More informationOn the Generalized Reid Inequality and the Numerical Radii
Applied Mathematical Sciences, Vol. 5, 2011, no. 9, 441-445 On the Generalized Reid Inequality and the Numerical Radii J. O. Bonyo 1, D. O. Adicka 2, J. O. Agure 3 1,3 Department of Mathematics and Applied
More informationPolynomial numerical hulls of order 3
Electronic Journal of Linear Algebra Volume 18 Volume 18 2009) Article 22 2009 Polynomial numerical hulls of order 3 Hamid Reza Afshin Mohammad Ali Mehrjoofard Abbas Salemi salemi@mail.uk.ac.ir Follow
More informationContents. Overview of Jensen s inequality Overview of the Kantorovich inequality. Goal of the lecture
Jadranka Mi ci c Hot () Jensen s inequality and its converses MIA2010 1 / 88 Contents Contents 1 Introduction Overview of Jensen s inequality Overview of the Kantorovich inequality Mond-Pečarić method
More informationChapter 1 Preliminaries
Chapter 1 Preliminaries 1.1 Conventions and Notations Throughout the book we use the following notations for standard sets of numbers: N the set {1, 2,...} of natural numbers Z the set of integers Q the
More informationGeneralized left and right Weyl spectra of upper triangular operator matrices
Electronic Journal of Linear Algebra Volume 32 Volume 32 (2017 Article 3 2017 Generalized left and right Weyl spectra of upper triangular operator matrices Guojun ai 3695946@163.com Dragana S. Cvetkovic-Ilic
More informationNorm inequalities related to the matrix geometric mean
isid/ms/2012/07 April 20, 2012 http://www.isid.ac.in/ statmath/eprints Norm inequalities related to the matrix geometric mean RAJENDRA BHATIA PRIYANKA GROVER Indian Statistical Institute, Delhi Centre
More informationNote on deleting a vertex and weak interlacing of the Laplacian spectrum
Electronic Journal of Linear Algebra Volume 16 Article 6 2007 Note on deleting a vertex and weak interlacing of the Laplacian spectrum Zvi Lotker zvilo@cse.bgu.ac.il Follow this and additional works at:
More informationInequalities for Convex and Non-Convex Functions
pplied Mathematical Sciences, Vol. 8, 204, no. 9, 5923-593 HIKRI Ltd, www.m-hikari.com http://dx.doi.org/0.2988/ams.204.4860 Inequalities for Convex and Non-Convex Functions Zlatko Pavić Mechanical Engineering
More informationInequalities for Modules and Unitary Invariant Norms
Int. J. Contemp. Math. Sciences Vol. 7 202 no. 36 77-783 Inequalities for Modules and Unitary Invariant Norms Loredana Ciurdariu Department of Mathematics Politehnica University of Timisoara P-ta. Victoriei
More informationOn cardinality of Pareto spectra
Electronic Journal of Linear Algebra Volume 22 Volume 22 (2011) Article 50 2011 On cardinality of Pareto spectra Alberto Seeger Jose Vicente-Perez Follow this and additional works at: http://repository.uwyo.edu/ela
More informationSEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES
SEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES S.S. DRAGOMIR Abstract. The main aim of this paper is to establish some connections that exist between the numerical
More informationProducts of commuting nilpotent operators
Electronic Journal of Linear Algebra Volume 16 Article 22 2007 Products of commuting nilpotent operators Damjana Kokol Bukovsek Tomaz Kosir tomazkosir@fmfuni-ljsi Nika Novak Polona Oblak Follow this and
More informationis a new metric on X, for reference, see [1, 3, 6]. Since x 1+x
THE TEACHING OF MATHEMATICS 016, Vol. XIX, No., pp. 68 75 STRICT MONOTONICITY OF NONNEGATIVE STRICTLY CONCAVE FUNCTION VANISHING AT THE ORIGIN Yuanhong Zhi Abstract. In this paper we prove that every nonnegative
More informationResearch Article On Some Improvements of the Jensen Inequality with Some Applications
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 323615, 15 pages doi:10.1155/2009/323615 Research Article On Some Improvements of the Jensen Inequality with
More informationPairs of matrices, one of which commutes with their commutator
Electronic Journal of Linear Algebra Volume 22 Volume 22 (2011) Article 38 2011 Pairs of matrices, one of which commutes with their commutator Gerald Bourgeois Follow this and additional works at: http://repository.uwyo.edu/ela
More informationLower bounds for the Estrada index
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012) Article 48 2012 Lower bounds for the Estrada index Yilun Shang shylmath@hotmail.com Follow this and additional works at: http://repository.uwyo.edu/ela
More informationY. J. Cho, M. Matić and J. Pečarić
Commun. Korean Math. Soc. 17 (2002), No. 4, pp. 583 592 INEQUALITIES OF HLAWKA S TYPE IN n-inner PRODUCT SPACES Y. J. Cho, M. Matić and J. Pečarić Abstract. In this paper, we give Hlawka s type inequalities
More informationSOME RESULTS ON 2-INNER PRODUCT SPACES
Novi Sad J. Math. Vol. 37, No. 2, 2007, 35-40 SOME RESULTS ON 2-INNER PRODUCT SPACES H. Mazaheri 1, R. Kazemi 1 Abstract. We onsider Riesz Theorem in the 2-inner product spaces and give some results in
More informationGENERALIZATION ON KANTOROVICH INEQUALITY
Journal of Mathematical Inequalities Volume 7, Number 3 (203), 57 522 doi:0.753/jmi-07-6 GENERALIZATION ON KANTOROVICH INEQUALITY MASATOSHI FUJII, HONGLIANG ZUO AND NAN CHENG (Communicated by J. I. Fujii)
More informationDihedral groups of automorphisms of compact Riemann surfaces of genus two
Electronic Journal of Linear Algebra Volume 26 Volume 26 (2013) Article 36 2013 Dihedral groups of automorphisms of compact Riemann surfaces of genus two Qingje Yang yangqj@ruc.edu.com Dan Yang Follow
More informationA note on estimates for the spectral radius of a nonnegative matrix
Electronic Journal of Linear Algebra Volume 13 Volume 13 (2005) Article 22 2005 A note on estimates for the spectral radius of a nonnegative matrix Shi-Ming Yang Ting-Zhu Huang tingzhuhuang@126com Follow
More informationA new upper bound for the eigenvalues of the continuous algebraic Riccati equation
Electronic Journal of Linear Algebra Volume 0 Volume 0 (00) Article 3 00 A new upper bound for the eigenvalues of the continuous algebraic Riccati equation Jianzhou Liu liujz@xtu.edu.cn Juan Zhang Yu Liu
More informationOperator norms of words formed from positivedefinite
Electronic Journal of Linear Algebra Volume 18 Volume 18 (009) Article 009 Operator norms of words formed from positivedefinite matrices Stephen W Drury drury@mathmcgillca Follow this and additional wors
More informationCHEBYSHEV INEQUALITIES AND SELF-DUAL CONES
CHEBYSHEV INEQUALITIES AND SELF-DUAL CONES ZDZISŁAW OTACHEL Dept. of Applied Mathematics and Computer Science University of Life Sciences in Lublin Akademicka 13, 20-950 Lublin, Poland EMail: zdzislaw.otachel@up.lublin.pl
More informationarxiv: v1 [math.fa] 1 Oct 2015
SOME RESULTS ON SINGULAR VALUE INEQUALITIES OF COMPACT OPERATORS IN HILBERT SPACE arxiv:1510.00114v1 math.fa 1 Oct 2015 A. TAGHAVI, V. DARVISH, H. M. NAZARI, S. S. DRAGOMIR Abstract. We prove several singular
More informationExtensions of interpolation between the arithmetic-geometric mean inequality for matrices
Bakherad et al. Journal of Inequalities and Applications 017) 017:09 DOI 10.1186/s13660-017-1485-x R E S E A R C H Open Access Extensions of interpolation between the arithmetic-geometric mean inequality
More informationL p Spaces and Convexity
L p Spaces and Convexity These notes largely follow the treatments in Royden, Real Analysis, and Rudin, Real & Complex Analysis. 1. Convex functions Let I R be an interval. For I open, we say a function
More informationJENSEN INEQUALITY IS A COMPLEMENT TO KANTOROVICH INEQUALITY
Scientiae Mathematicae Japonicae Online, e-005, 91 97 91 JENSEN NEQUALTY S A COMPLEMENT TO KANTOROVCH NEQUALTY MASATOSH FUJ AND MASAHRO NAKAMURA Received December 8, 004; revised January 11, 005 Dedicated
More informationSingular Value Inequalities for Real and Imaginary Parts of Matrices
Filomat 3:1 16, 63 69 DOI 1.98/FIL16163C Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Singular Value Inequalities for Real Imaginary
More informationJensen s inequality for nonconvex functions
Mathematical Communications 9(004, 119-14 119 Jensen s inequality for nonconvex functions Sanjo Zlobec Abstract. Jensen s inequality is formulated for convexifiable (generally nonconvex functions. Key
More informationELA MAPS PRESERVING GENERAL MEANS OF POSITIVE OPERATORS
MAPS PRESERVING GENERAL MEANS OF POSITIVE OPERATORS LAJOS MOLNÁR Abstract. Under some mild conditions, the general form of bijective transformations of the set of all positive linear operators on a Hilbert
More informationOn Projection of a Positive Definite Matrix on a Cone of Nonnegative Definite Toeplitz Matrices
Electronic Journal of Linear Algebra Volume 33 Volume 33: Special Issue for the International Conference on Matrix Analysis and its Applications, MAT TRIAD 2017 Article 8 2018 On Proection of a Positive
More informationarxiv: v1 [math.fa] 13 Dec 2011
NON-COMMUTATIVE CALLEBAUT INEQUALITY arxiv:.3003v [ath.fa] 3 Dec 0 MOHAMMAD SAL MOSLEHIAN, JAGJIT SINGH MATHARU AND JASPAL SINGH AUJLA 3 Abstract. We present an operator version of the Callebaut inequality
More informationProof of Atiyah's conjecture for two special types of configurations
Electronic Journal of Linear Algebra Volume 9 Volume 9 (2002) Article 14 2002 Proof of Atiyah's conjecture for two special types of configurations Dragomir Z. Djokovic dragomir@herod.uwaterloo.ca Follow
More informationAvailable online at J. Math. Comput. Sci. 2 (2012), No. 3, ISSN:
Available olie at http://scik.org J. Math. Comput. Sci. 2 (202, No. 3, 656-672 ISSN: 927-5307 ON PARAMETER DEPENDENT REFINEMENT OF DISCRETE JENSEN S INEQUALITY FOR OPERATOR CONVEX FUNCTIONS L. HORVÁTH,
More informationGeneralized Numerical Radius Inequalities for Operator Matrices
International Mathematical Forum, Vol. 6, 011, no. 48, 379-385 Generalized Numerical Radius Inequalities for Operator Matrices Wathiq Bani-Domi Department of Mathematics Yarmouk University, Irbed, Jordan
More informationAvailable online at J. Math. Comput. Sci. 4 (2014), No. 1, 1-9 ISSN:
Available online at http://scik.org J. Math. Comput. Sci. 4 (2014), No. 1, 1-9 ISSN: 1927-5307 ON -n-paranormal OPERATORS ON BANACH SPACES MUNEO CHŌ 1, KÔTARÔ TANAHASHI 2 1 Department of Mathematics, Kanagawa
More informationConvex Optimization Notes
Convex Optimization Notes Jonathan Siegel January 2017 1 Convex Analysis This section is devoted to the study of convex functions f : B R {+ } and convex sets U B, for B a Banach space. The case of B =
More informationGeneralized Simpson-like Type Integral Inequalities for Differentiable Convex Functions via Riemann-Liouville Integrals
International Journal of Mathematical Analysis Vol. 9, 15, no. 16, 755-766 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/ijma.15.534 Generalized Simpson-like Type Integral Ineualities for Differentiable
More informationMultiplication Operators with Closed Range in Operator Algebras
J. Ana. Num. Theor. 1, No. 1, 1-5 (2013) 1 Journal of Analysis & Number Theory An International Journal Multiplication Operators with Closed Range in Operator Algebras P. Sam Johnson Department of Mathematical
More informationA new parallel polynomial division by a separable polynomial via hermite interpolation with applications, pp
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012 Article 54 2012 A new parallel polynomial division by a separable polynomial via hermite interpolation with applications, pp 770-781 Aristides
More informationPartial isometries and EP elements in rings with involution
Electronic Journal of Linear Algebra Volume 18 Volume 18 (2009) Article 55 2009 Partial isometries and EP elements in rings with involution Dijana Mosic dragan@pmf.ni.ac.yu Dragan S. Djordjevic Follow
More informationInterior points of the completely positive cone
Electronic Journal of Linear Algebra Volume 17 Volume 17 (2008) Article 5 2008 Interior points of the completely positive cone Mirjam Duer duer@mathematik.tu-darmstadt.de Georg Still Follow this and additional
More informationOn Some Estimates of the Remainder in Taylor s Formula
Journal of Mathematical Analysis and Applications 263, 246 263 (2) doi:.6/jmaa.2.7622, available online at http://www.idealibrary.com on On Some Estimates of the Remainder in Taylor s Formula G. A. Anastassiou
More informationarxiv: v1 [math.fa] 6 Nov 2015
CARTESIAN DECOMPOSITION AND NUMERICAL RADIUS INEQUALITIES FUAD KITTANEH, MOHAMMAD SAL MOSLEHIAN AND TAKEAKI YAMAZAKI 3 arxiv:5.0094v [math.fa] 6 Nov 05 Abstract. We show that if T = H + ik is the Cartesian
More information