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Let A and B be two positive operators with for positive real numbers be an operator mean and be the adjoint mean of If and is a positive unital linear map, then where and is the Kantorovich constant. In addition, for [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a comprehensive study of operator m-convex functions.Let m ∈ [ 0 , 1 ] {m\in[0,1]} , and J = [ 0 , b ] {J=[0,b]} for some b ∈ ℝ {b\in\mathbb{R}} or J = [ 0 , ∞ ) {J=[0,\infty)}.A continuous function φ : J → ℝ {\varphi\colon J\to\mathbb{R}} is called operator m-convex if for any t ∈ [ 0 , 1 ] {t\in[0,1]} and any self-adjoint operators A , B ∈ 𝔹 ⁢ ( ℋ ) {A,B\in\mathbb{B}({\mathscr{H}})} , whose spectra are contained in J, we have φ ⁢ ( t ⁢ A + m ⁢ ( 1 - t ) ⁢ B ) ≤ t ⁢ φ ⁢ ( A ) + m ⁢ ( 1 - t ) ⁢ φ ⁢ ( B ) {\varphi(tA+m(1-t)B)\leq t\varphi(A)+m(1-t)\varphi(B)}.We first generalize the celebrated Jensen inequality for continuous m-convex functions and Hilbert space operators and then use suitable weight functions to give some weighted refinements.Introducing the notion of operator m-convexity, we extend the Choi–Davis–Jensen inequality for operator m-convex functions.We also present an operator version of the Jensen–Mercer inequality for m-convex functions and generalize this inequality for operator m-convex functions involving continuous fields of operators and unital fields of positive linear mappings.Employing the Jensen–Mercer operator inequality for operator m-convex functions, we construct the m-Jensen operator functional and obtain an upper bound for it. [ABSTRACT FROM AUTHOR]

We present some generalizations of quantum information inequalities involving tracial positive linear maps between C ⁎ -algebras. Among several results, we establish a noncommutative Heisenberg uncertainty relation. More precisely, we show that if Φ : A → B is a tracial positive linear map between C ⁎ -algebras, ρ ∈ A is a Φ-density element and A , B are self-adjoint operators of A such that sp ( -i ρ 1 2 [ A , B ] ρ 1 2 ) ⊆ [ m , M ] for some scalers 0 < m < M , then under some conditions (0.1) V ρ , Φ ( A ) ♯ V ρ , Φ ( B ) ≥ 1 2 K m , M ( ρ [ A , B ] ) | Φ ( ρ [ A , B ] ) | , where K m , M ( ρ [ A , B ] ) is the Kantorovich constant of the operator -i ρ 1 2 [ A , B ] ρ 1 2 and V ρ , Φ ( X ) is the generalized variance of X . In addition, we use some arguments differing from the scalar theory to present some inequalities related to the generalized correlation and the generalized Wigner–Yanase–Dyson skew information. [ABSTRACT FROM AUTHOR]

Using a refinement of the classical Young inequality, we refine some inequalities of operators including the function ω p , where ω p is defined for p ⩾ 1 and operators T 1 , … , T n ∈ B ( H ) by ω p ( T 1 , … , T n ) : = sup ‖ x ‖ = 1 ⁡ ( ∑ i = 1 n | 〈 T i x , x 〉 | p ) 1 p . Among other things, we show that if T 1 , … , T n ∈ B ( H ) and p ≥ q ≥ 1 with 1 p + 1 q = 1 , then 1 n ‖ ∑ i = 1 n T i ‖ 2 ≤ ω p ( | T 1 | , … , | T n | ) ω q ( | T 1 ⁎ | , … , | T n ⁎ | ) 1 n ‖ ∑ i = 1 n T i ‖ 2 ≤ 1 p ‖ ∑ i = 1 n | T i | p ‖ + 1 q ‖ ∑ i = 1 n | T i ⁎ | q ‖ − inf ‖ x ‖ = ‖ y ‖ = 1 ⁡ δ ( x , y ) , where δ ( x , y ) = 1 p ( ∑ i = 1 n 〈 | T i | x , x 〉 p − ∑ i = 1 n 〈 | T i ⁎ | y , y 〉 q ) 2 . [ABSTRACT FROM AUTHOR]

In this paper, motivated by perturbation theory of operators, we present some upper bounds for ⦀ f ( A ) X g ( B ) + X ⦀ in terms of ⦀ | A X B | + | X | ⦀ and ⦀ f ( A ) X g ( B ) − X ⦀ in terms of ⦀ | A X | + | X B | ⦀ , where A , B are G 1 operators, ⦀ ⋅ ⦀ is a unitarily invariant norm and f , g are certain analytic functions. Further, we find some new upper bounds for the Schatten 2-norm of f ( A ) X ± X g ( B ) . Several special cases are discussed as well. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a new class of functions, which is developed via the Hermite–Hadamard inequality for convex functions. The well-known one-one correspondence between the class of operator monotone functions and operator connections declares that the obtained class represents the weighted logarithmic means. We shall also consider weighted identric mean and some relationships between various operator means. Among many things, we extended the weighted arithmetic–geometric operator mean inequality asandinvolving the considered operator means. [ABSTRACT FROM PUBLISHER]

In this paper, we obtain some refinements of the known operator inequalities for the p -Schatten class. In addition, we obtain an approach to the inequalities conjectured by Audenaert and Kittaneh for the p -Schatten class. [ABSTRACT FROM AUTHOR]

The symmetric doubly stochastic inverse spectral problem is the problem of determining necessary and sufficient conditions for a real-tuple to be the spectrum of ansymmetric doubly stochastic matrix. For, this problem remains open though many partial results are known. In this note, we present a new family of necessary conditions for this problem using some matrix trace inequalities. In addition, we prove that this family of new inequalities sharpen the existing known necessary conditions for the inverse spectral problem of nonnegative matrices. Finally, we prove that these necessary conditions are not sufficient for the case. [ABSTRACT FROM AUTHOR]

The Callebaut inequality says thatwhereare positive invertible operators, andandare an operator mean and its dual in the sense of Kabo and Ando, respectively. In this paper we employ the Mond–Pečarić method as well as some operator techniques to establish a complementary inequality to the above one under mild conditions. We also present some refinements of a Callebaut-type inequality involving the weighted geometric mean and Hadamard products of Hilbert space operators. [ABSTRACT FROM AUTHOR]

We show that if T = H + i K is the Cartesian decomposition of T ∈ B ( H ) , then for α , β ∈ R , sup α 2 + β 2 = 1 ⁡ ‖ α H + β K ‖ = w ( T ) . We then apply it to prove that if A , B , X ∈ B ( H ) and 0 ≤ m I ≤ X , then m ‖ Re ( A ) − Re ( B ) ‖ ≤ w ( Re ( A ) X − X Re ( B ) ) ≤ 1 2 sup θ ∈ R ⁡ ‖ ( A X − X B ) + e i θ ( X A − B X ) ‖ ≤ ‖ A X − X B ‖ + ‖ X A − B X ‖ 2 , where Re ( T ) denotes the real part of an operator T . A refinement of the triangle inequality is also shown. [ABSTRACT FROM AUTHOR]

We generalize several inequalities involving powers of the numerical radius for product of two operators acting on a Hilbert space. For any A , B , X ∈ B ( H ) such that A , B are positive, we establish some numerical radius inequalities for A α X B α and A α X B 1 − α ( 0 ≤ α ≤ 1 ) and Heinz means under mild conditions. [ABSTRACT FROM AUTHOR]

We establish several operator extensions of the Chebyshev inequality. The main version deals with the Hadamard product of Hilbert space operators. More precisely, we prove that if A is a C\*-algebra, T is a compact Hausdorff space equipped with a Radon measure µ, α:T→[0,+∞) is a measurable function and (At)t∈T, (Bt)t∈T are suitable continuous fields of operators in A having the synchronous Hadamard property, then Tα(s)dµ(s) Tα(t)(At Bt)dµ(t)≥(Tα(t)Atdµ(t)) (Tα(s)Bsdµ(s)). We apply states on C\*-algebras to obtain some versions related to synchronous functions. We also present some Chebyshev type inequalities involving the singular values of positive n×n matrices. Several applications are given as well. [ABSTRACT FROM AUTHOR]

Abstract: The celebrated Heinz inequality asserts that for , , every unitarily invariant norm and . In this paper, we present several improvement of the Heinz inequality by using the convexity of the function , some integration techniques and various refinements of the Hermite–Hadamard inequality. In the setting of matrices we prove that for real numbers α, β. [Copyright &y& Elsevier]

Abstract: We extend the celebrated Löwner–Heinz inequality by showing that if are Hilbert space operators such that , thenfor each . As an application we prove that [Copyright &y& Elsevier]

There are several inequalities in the literature carrying the name of Ky Fan. We survey these well-known Ky Fan inequalities and some other significant inequalities generalized by Ky Fan and review some of their recent developments. [ABSTRACT FROM AUTHOR]

Abstract: We present an operator version of the Callebaut inequality involving the interpolation paths and apply it to the weighted operator geometric means. We also establish a matrix version of the Callebaut inequality and as a consequence obtain an inequality including the Hadamard product of matrices. [Copyright &y& Elsevier]

Abstract: We prove some refinements of an inequality due to X. Zhan in an arbitrary complex Hilbert space by using some results on the Heinz inequality. We present several related inequalities as well as new variants of the Corach–Porta–Recht inequality. We also characterize the class of operators satisfying under certain conditions. [Copyright &y& Elsevier]

Some inequalities of Jensen type for Q-class functions are proved. More precisely, a refinement of the inequality f((1/P)∑ ni=1pixi)≤P∑ ni=1(f(xi)/pi) is given in which p1,…,pn are positive numbers, P=∑ ni=1pi and f is a Q-class function. The notion of the jointly Q-class function is introduced and some Jensen type inequalities for these functions are proved. Some Ostrowski and Hermite–Hadamard type inequalities related to Q-class functions are presented as well. [ABSTRACT FROM PUBLISHER]

Abstract: For a normed linear space and we characterize all -tuples for which the generalized triangle inequality of the second type holds for any . We also characterize for which the reverse of the inequality above holds. [Copyright &y& Elsevier]

Abstract: Let be the Schatten p-class for . Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: if and are two sets of operators in, then In this paper, we give generalizations of this as pairs of inequalities for Schatten p-norms, which hold for certain values of p and reduce to the equality above for . Moreover, we present some related inequalities for three sets of operators. [Copyright &y& Elsevier]

Abstract: We present a weak majorization inequality and apply it to prove eigenvalue and unitarily invariant norm extensions of a version of the Bohr’s inequality due to Vasić and Kečkić. [Copyright &y& Elsevier]

Abstract: We establish several operator versions of the classical Aczél inequality. One of operator versions deals with the weighted operator geometric mean and another is related to the positive sesquilinear forms. Some applications including the unital positive linear maps on -algebras and the unitarily invariant norms on matrices are presented. [Copyright &y& Elsevier]

Abstract: Let be a unital -algebra and let be a unital n-positive linear map between -algebras for some . We show thatfor all operators , where denotes the operator norm distance of C from the scalar operators. [Copyright &y& Elsevier]

Abstract: Let be a -algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let be a continuous field of operators in such that the function is norm continuous on T and the function is integrable. Then the following equality including Bouchner integrals holds This equality is related both to the notion of variance in statistics and to a characterization of inner product spaces. With this operator equality, we present some uniform norm and Schatten p-norm inequalities. [Copyright &y& Elsevier]

Abstract: We study the Cauchy–Schwarz and some related inequalities in a semi-inner product module over a -algebra . The key idea is to consider a semi-inner product -module as a semi-inner product -module with respect to another semi-inner product. In this way, we improve some inequalities such as the Ostrowski inequality and an inequality related to the Gram matrix. The induced semi-inner products are also related to the notion of covariance and variance. Furthermore, we obtain a sequence of nested inequalities that emerges from the Cauchy–Schwarz inequality. As a consequence, we derive some interesting operator-theoretical corollaries. In particular, we show that the sequence arising from our construction, when applied to a positive invertible element of a -algebra, converges to its inverse. [Copyright &y& Elsevier]