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We improve the generalized numerical radius version of the Heinz means inequalities by employing convexity and the Hermit-Hadamard inequality. Moreover, we present novel proofs for the generalized numerical radius of Heinz-type inequalities. [ABSTRACT FROM AUTHOR]

For positive definite matrices A and B, the Kubo-Ando matrix power mean is defined as \[ P_\mu(p, A, B) = A^{1/2}\left(\frac{1+(A^{-1/2}BA^{-1/2})^p}{2}\right)^{1/p} A^{1/2}\quad (p \ge 0). \] P μ (p , A , B) = A 1 / 2 (1 + (A − 1 / 2 B A − 1 / 2 ) p 2) 1 / p A 1 / 2 (p ≥ 0). In this paper, for $ 0\le p \le 1 \le q $ 0 ≤ p ≤ 1 ≤ q , we show that if one of the following inequalities \[ f(P_\mu(p, A, B)) \le f(P_\mu(1, A, B)) \le f(P_\mu(q, A, B)) \] f (P μ (p , A , B)) ≤ f (P μ (1 , A , B)) ≤ f (P μ (q , A , B)) holds for any positive definite matrices A and B, then the function f is operator monotone on $ (0, \infty). $ (0 , ∞). We also study the inverse problem for non-Kubo-Ando matrix power means with the powers 1/2 and 2. As a consequence, we establish new charaterizations of operator monotone functions with the non-Kubo-Ando matrix power means. [ABSTRACT FROM AUTHOR]

Let S α (0 ≤ α < π 2) stand for the set of all complex sector matrices and σ 1 , σ 2 be two operator means satisfying σ 1 ≤ σ 2. Except some other assertions, it is also shown that for A , B ∈ S α , ℜ (A σ 1 B) ≤ sec 2 α ℜ (A σ 2 B) and ℜ (A σ 2 B) - 1 ≤ sec 2 α ℜ (A σ 1 B) - 1. In addition, if σ i ∗ ≤ σ i , for i = 1 , 2 and Φ is a unital positive linear map, then Φ ℜ (A σ 1 B) - 1 ≤ sec 2 α ℜ (Φ (A - 1) σ 2 Φ (B - 1)). [ABSTRACT FROM AUTHOR]

In this article, we introduce the definition of the weighted numerical radius ω ν (T) of a Hilbert space operator T, and present.many interesting properties of this newly defined notion. In particular, we find some bounds for ω ν and study its relation with the numerical radius ω (·) . Some bounds for ω ν will be matched with known bounds for ω . [ABSTRACT FROM AUTHOR]

The main goal of this paper is to discuss the recent advancements of matrix means from positive matrices to accretive matrices in a more general setting. In particular, we present the general form governing the well established definition of geometric mean, then we define arbitrary matrix means and functional calculus for accretive matrices. Applications of this new discussion involve generalizations of known inequalities from the setting of positive matrices to that of accretive matrices. This includes the arithmetic-harmonic mean comparisons, monotonicity of matrix means, Ando's inequality, Choi's inequality, Ando–Zhan subadditive inequality and much more. [ABSTRACT FROM AUTHOR]

The numerical radius of a matrix is a scalar quantity that has many applications in the study of matrix analysis. Due to the difficulty in computing the numerical radius, inequalities bounding it have received a considerable attention in the literature. In this article, we present many new inequalities for the numerical radius of accretive matrices. The importance of this study is the presence of a new approach that treats a specific class of matrices, namely the accretive ones. While some of these inequalities can be considered as refinements of other existing ones, others present new insight to some known results for positive matrices. [ABSTRACT FROM AUTHOR]

Strict monotonicity of the spectral radii of bounded, positive, ordered linear operators is investigated. It is well-known that under reasonable assumptions, the spectral radii of two ordered positive operators enjoy a non-strict inequality. It is also well-known that a “strict” inequality between operators does not imply strict monotonicity of the spectral radii in general—some additional structure is required. We present a number of sufficient conditions on both the cone and the operators for such a strict ordering to hold which generalise known results in the literature, and have utility in comparison arguments, ubiquitous in positive systems theory. [ABSTRACT FROM AUTHOR]

If σ is a symmetric mean and f is an operator monotone function on [ 0 , ∞ ) , then f ( 2 ( A − 1 + B − 1 ) − 1 ) ≤ f ( A σ B ) ≤ f ( ( A + B ) / 2 ) . Conversely, Ando and Hiai showed that if f is a function that satisfies either one of these inequalities for all positive operators A and B and a symmetric mean different than the arithmetic and the harmonic mean, then the function is operator monotone. In this paper, we show that the arithmetic and the harmonic means can be replaced by the geometric mean to obtain similar characterizations. Moreover, we give characterizations of operator monotone functions using self-adjoint means and general means subject to a constraint due to Kubo and Ando. [ABSTRACT FROM AUTHOR]

Polynomial interpolation has been used for centuries to find approximate values of a function, then to find related quantities like integrals and derivatives. In this article, we present a completely new application of polynomial interpolation. More precisely, we implement an inductive quadratic interpolation formula to obtain refinements and reverses of Young's inequality for numbers and matrices. The matrices applications include determinants and ordering inequalities. [ABSTRACT FROM AUTHOR]

In this article, we prove several multi-term refinements of Young type inequalities for both real numbers and operators improving several known results. Among other results, we prove that for all 0 ≤ ν ≤ 1 and each N ∈ N , A # ν B + ∑ j = 1 N s j ( ν ) ( A # α j ( ν ) B + A # 2 1 − j + α j ( ν ) B − 2 A # 2 − j + α j ( ν ) B ) ≤ A ∇ ν B , for the positive operators A and B , where s j ( ν ) and α j ( ν ) are certain functions. Moreover, some new Heinz type inequalities involving the Hilbert–Schmidt norm are established. [ABSTRACT FROM AUTHOR]

In this article we present some mean inequalities for convex functions that lead to some generalized inequalities treating the arithmetic, geometric and harmonic means for numbers and matrices. Our first main inequality will be ( ν τ ) λ ≤ ( ( 1 − ν ) f ( 0 ) + ν f ( 1 ) ) λ − f λ ( ν ) ( ( 1 − τ ) f ( 0 ) + τ f ( 1 ) ) λ − f λ ( τ ) ≤ ( 1 − ν 1 − τ ) λ , for the convex function f , when λ ≥ 1 and 0 < ν ≤ τ < 1 . Moreover, when λ = 1 , the inequality will be valid for operator convex functions. Then by selecting an appropriate convex function, we obtain certain matrix inequalities. In particular, we obtain several mixed mean inequalities for operators using real and operator convexity. Our discussion will lead to new multiplicative refinements and reverses of the Heinz and Hölder inequalities for matrices, new and refined trace and determinant inequalities. The significance of this work is its general treatment, where convexity is the only needed property. [ABSTRACT FROM AUTHOR]

In this article, we interpolate the well-known Young’s inequality for numbers and matrices, when equipped with the Hilbert–Schmidt norm, then present the corresponding interpolations of recent refinements in the literature. As an application of these interpolated versions, we study the monotonicity these interpolations obey. [ABSTRACT FROM PUBLISHER]

In this article we interpolate the well known Young and Heinz inequalities for unitarily invariant norms, and some of their known refinements. Then we prove new interpolated refinements. In the end, we use this interpolation idea to prove a hidden monotonicity behavior these inequalities obey. [ABSTRACT FROM AUTHOR]