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In this paper, we introduce operator geodesically convex and operator convex-log functions and characterize some properties of them. Then we apply these classes of functions to present several operator Azcél and Minkowski-type inequalities extending some known results. The concavity counterparts are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we firstly establish new numerical radius inequalities which refine a result of Kittaneh in [Studia Math. 168, 73–80 (2005)], then present some numerical radius inequalities involving non-negative increasing convex functions for n × n operator matrices, which generalize the related results of Shebrawi in [Linear Algebra Appl. 523(15), 1–12 (2017)]. [ABSTRACT FROM AUTHOR]

A continuous function f : I × I → R is called operator Schur convex, if f is symmetric, namely f x , y = f y , x for all x, y ∈ I and f t A + 1 − t B , t B + 1 − t A ≤ f A , B in the operator order, for all A , B ∈ S A I H × S A I H and t ∈ 0 , 1 , where S A I H is the convex set of all selfadjoint operators on Hilbert space H with spectra in I. In this paper we investigate the main properties of such functions, establish some integral inequalities of Hermite–Hadamard, Čebyšev and Grüss' type and give some general classes of examples of operator Schur-convex functions. [ABSTRACT FROM AUTHOR]

Letpbe a positive number andha function onsatisfyingfor any. A non-negative continuous functionfonis said to beoperator (p, h)-convexif holds for all positive semidefinite matricesA, Bof ordernwith spectra inK, and for any. In this paper, we study properties of operator (p, h)-convex functions and prove the Jensen, Hansen–Pedersen type inequalities for them. We also give some equivalent conditions for a function to become an operator (p, h)-convex. In applications, we obtain Choi–Davis–Jensen type inequality for operator (p, h)-convex functions and a relation between operator (p, h)-convex functions with operator monotone functions. [ABSTRACT FROM AUTHOR]

In this paper, the weighted norm inequalities of the following special type of integral operators are investigated: whereor. In particular, those established by E. T. Copson, Hardy-Littlewood-Pólya, S. Lai, N. Levinson, E. R. Love, B. G. Pachpatte and Yang-Hwang will be derived. [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]

LetBbe andoubly substochastic matrix and letsbe the sum of all entries ofB. In this paper, we show thatBhas a sub-defect ofk,  which can be computed by taking the ceiling ofif and only if there exists andoubly stochastic extension containingBas a submatrix andkminimal. We also propose a procedure constructing a minimal completion ofB, and then express it as a convex combination of partial permutation matrices. [ABSTRACT FROM PUBLISHER]

Let A and B be $ n\times n $ n × n positive semidefinite matrices, $ f:[0,\infty)\rightarrow \lbrack 0,\infty) $ f : [ 0 , ∞) → [ 0 , ∞) be an increasing convex function with $ f\left (0\right) =0 $ f (0) = 0 , and let $ \Phi : \mathbb {M}_{n}(\mathbb {C})\rightarrow \mathbb {M}_{n}(\mathbb {C}) $ Φ : M n (C) → M n (C) be a positive unital linear map. It is shown that $$\begin{align*} & \prod_{j=1}^{k} s_{j}\left(f^{4}\left(\Phi \left(\frac{A+B}{2}\right) \right) \right)\\ & \quad \leq 2^{-4} \prod_{j=1}^{k} \left(s_{j}\left(\Phi^{2}\left(f^{2}\left(A\right) \right) +\Phi \left(f^{2}\left(A\right) \right) \right) +\left\Vert \Phi \left(f^{2}\left(B\right) \right) \right\Vert +1\right) \\ & \quad\quad \times \left(s_{j}\left(\Phi^{2}\left(f^{2}\left(B\right) \right) +\Phi \left(f^{2}\left(B\right) \right) \right) +\left\Vert \Phi \left(f^{2}\left(A\right) \right) \right\Vert +1\right) \end{align*}$$ ∏ j = 1 k s j (f 4 (Φ (A + B 2))) ≤ 2 − 4 ∏ j = 1 k (s j (Φ 2 (f 2 (A)) + Φ (f 2 (A))) + ‖ Φ (f 2 (B)) ‖ + 1) × (s j (Φ 2 (f 2 (B)) + Φ (f 2 (B))) + ‖ Φ (f 2 (A)) ‖ + 1) for $ k=1,2,\ldots,n $ k = 1 , 2 , ... , n , where $ s_{j}\left (T\right) $ s j (T) and $ \left \Vert T\right \Vert $ ‖ T ‖ are the $ j{\rm th} $ j th singular value and the spectral norm of T, respectively. Applications of our results to sectorial matrices are given. [ABSTRACT FROM AUTHOR]

Let Ai, P, i = 1, ..., m, be n by n complex matrices such that each Ai is positive definite. It is shown, among other inequalities, that If 0 < aj ≤ sn(Aj), C m = ∑ j = 1 m (∑ i = 1 , i ≠ j m A i ) a j and P is Hermitian, then for every unitarily invariant norm ∥ |. ∥ | , ∥ | C m P + P C m ∥ | ≥ α ∥ | P ∥ | , where α = max 2 m − 1 , m 1 + min a 1 2 , ... , a m 2 . If s1(Aj) ≤ bj, ∑ i = 1 m b i = 1 and D m = ∑ j = 1 m (∑ i = 1 , i ≠ j m A i ) b j , then for every unitarily invariant norm ∥ |. ∥ | , 2 m P ≥ ∥ | D m P + P D m ∥ |. [ABSTRACT FROM AUTHOR]

Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a 'supertropical trigonometry' and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars in the ray space R a y (V). In particular, these functions induce a partition of R a y (V) into convex sets, and thereby a finer convex analysis which includes the notions of median, minima, glens, and polars. [ABSTRACT FROM AUTHOR]

We present some unitarily invariant norm and Schatten p-norm inequalities relevant to accretive-dissipative matrices involving convex and concave functions. In particular, we show that if T = T i j 2 × 2 is an accretive-dissipative block matrix with T i j ∈ M n (C) , i , j = 1 , 2 , and f is an increasing convex function on [ 0 , ∞) with f (0) = 0 , then 2 f | T | 2 u ≤ ∥ f (2 | T 11 |) ∥ u + ∥ f (2 | T 22 |) ∥ u , for every unitarily invariant norm ∥ ⋅ ∥ u . We also extract several inequalities for accretive-dissipative n × n operator matrices. The obtained inequalities extend some known results. [ABSTRACT FROM AUTHOR]

In this article, matrix-valued functions nonregular on the convex hull of the spectrum are considered. The logarithm, root and meromorphic functions of various matrices are examples of such functions. Here sharp norm estimates are established for the considered matrix-valued functions. Applications of the obtained estimates to the periodic problem and two-points boundary value problem for vector differential equations are also discussed. [ABSTRACT FROM PUBLISHER]

We approximate a class of operator convex functions via some numerical results which were obtained by Guessab, and determine the error of the approximation. Finally we represent a characterization of special operator integration, by which we obtain error bound for left-hand side inequality of Hermite-Hadamard inequality. [ABSTRACT FROM AUTHOR]

In a recent work of the authors, we showed some general inequalities governing numerical radius inequalities using convex functions. In this article, we present results that complement the aforementioned inequalities. In particular, the new versions can be looked at as refined and generalized forms of some well-known numerical radius inequalities. Among many other results, we show that f A ∗ A + A A ∗ 4 ≤ ∫ 0 1 f 1 − t B 2 + t C 2 d t ≤ f w 2 A , where A is a bounded linear operator on a Hilbert space having the Cartesian decomposition A = B + iC and f : [ 0 , ∞) → [ 0 , ∞) is an increasing operator convex function. This result, for example, extends and refines a celebrated result by Kittaneh. [ABSTRACT FROM AUTHOR]

Let be bounded linear operators on a complex separable Hilbert space and let be positive real numbers such that and. Among other results, it is shown that (a) If f is a non-negative function on such that and is convex, then for every unitarily invariant norm, for. (b) If f is a non-negative function on such that is concave, then for every unitarily invariant norm, for. Here [ABSTRACT FROM AUTHOR]

Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, if any, being positive. Invertibility of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A having positive principal minors (i.e. being a P-matrix). These results are extended by considering convex combinations of higher powers of A and of their rows. The invertibility of matrices in these convex hulls is associated with the eigenvalues of A lying in open sectors of the right-half plane and provides a general context for the theory of matrices with P-matrix powers. [ABSTRACT FROM AUTHOR]

Letfbe a continuous convex function on an intervalJandbe Hilbert spaces. Letbe self-adjoint operators with spectra in [m, M] for some scalarsandbe self-adjoint operators with spectra in. Ifis ak-tuple andis an-tuple of positive linear mappings fromintowith,and, then we prove the inequalities As applications, we prove Jensen’s inequality for operators without operator convexity and obtain some operator inequalities of Jensen type. [ABSTRACT FROM AUTHOR]

We construct a large class of indecomposable positive linear maps from thematrix algebra into thematrix algebra, which generate exposed extreme rays of the convex cone of all positive maps. We show that extreme points of the dual faces for separable states arising from these maps are parametrized by the Riemann sphere, and the convex hulls of the extreme points arising from a circle parallel to the equator have exactly the same properties with the convex hull of the trigonometric moment curve studied from combinatorial topology. Any interior points of the dual faces areboundary separable states with full ranks. We exhibit concrete examples of such states. [ABSTRACT FROM PUBLISHER]

We investigate the non-commutative-divergence functionalof operator log-convex functions. In particular, we show that a continuous functionis operator log-convex if and only if for all continuous fieldsandof strictly positive operators. Moreover, we define operator-convex functions and investigate some of its properties. In particular, we show that this class contains operator convex functions and operator log-convex functions. [ABSTRACT FROM AUTHOR]

Sub-additive and super-additive inequalities for concave and convex functions have been generalized to the case of matrices by several authors over a period of time. These lead to some interesting inequalities for matrices, which in some cases coincide with, and in other cases are at variance with the corresponding inequalities for real numbers. We survey some of these matrix inequalities and do further investigations into these. We introduce the novel notion of dominated majorization between the spectra of two Hermitian matrices B and C, dominated by a third Hermitian matrix A. Based on an explicit formula for the gradient of the sum of the k largest eigenvalues of a Hermitian matrix, we show that under certain conditions dominated majorization reduces to a linear majorization-like relation between the diagonal elements of B and C in a certain basis. We use this notion as a tool to give new, elementary proofs for the sub-additivity inequality for non-negative concave functions first proved by Bourin and Uchiyama and the corresponding super-additivity inequality for non-negative convex functions first proven by Kosem. Finally, we present counterexamples to some conjectures that Ando's inequality for operator convex functions could more generally hold, e.g. for ordinary convex, non-negative functions. [ABSTRACT FROM AUTHOR]