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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]

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]

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]