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In this paper, we estimate the difference between the Hadamard product and the Karcher mean of n positive invertible operators on the Hilbert space in terms of the Specht ratio and the Kantorovich constant. Also, we improve the obtained inequalities in the case of n = 2. Moreover, we give ratio inequalities of the operator power means by the Hadamard product. [ABSTRACT FROM AUTHOR]

In this paper, extesions of some inequalities involving matrix means and sector matrices are considered. Among other results, we prove that if two sector matrices A and B satisfy 0 < mIn ≥; ℜ(A),ℜ(B) ≥ MIn, then Φp (ℜ(Aσ B)) ≥ sec2p(θ)Kp(h)Φp(ℜ(Bσ ⊥A)), (0 ≥ p ≥ 2) for every unital positive linear map Φ and arbitrary mean σ, where K(h) := (M+m)² 4Mm is the Kantorovich constant with h := Mm. In addition, we present some norm, numerical radius and determinantal inequalities for sector matrices. [ABSTRACT FROM AUTHOR]

The ordering betweenWigner-Yanase-Dyson function and logarithmic mean is known. Also bounds for logarithmic mean are known. In this paper, we give two reverse inequalities for Wigner-Yanase-Dyson function and logarithmic mean. We also compare the obtained results with the known bounds of the logarithmic mean. Finally, we give operator inequalities based on the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we give some new improvements and reverse improvements of Young type inequalities. The conclusion proved by Yang and Wang [J. Math. Inequal., 17 (2023), 205-217] involved the monotonicity of ..., where ... and ... . This article demonstrates the monotonicity of ..., where ... and ... . And this implies a main conclusion as follows: ..., where ... and ... . Furthermore, we can get some related results about operator, Hilbert-Schmidt norm, trace norm by these scalars results. [ABSTRACT FROM AUTHOR]

The main goal of this article is to present new inequalities for the spectral geometric mean $ A\natural _t B $ A ♮ t B of two positive definite operators A, B on a Hilbert space. The obtained results complement many known inequalities for the geometric mean $ A\sharp _t B $ A ♯ t B. In particular, explicit comparisons between $ A\natural _t B $ A ♮ t B and $ A\sharp _t B $ A ♯ t B are given, with applications towards Ando-type inequalities and Ando-Hiai inequalities for $ A\natural _t B $ A ♮ t B and some other consequences. [ABSTRACT FROM AUTHOR]

This article introduces new Young-type inequalities, leveraging the Kantorovich constant, by refining the original inequality. In addition, we present a range of norm-based inequalities applicable to positive semidefinite matrices, such as the Hilbert-Schmidt norm and the trace norm. The importance of these results lies in their dual significance: they hold inherent value on their own, and they also extend and build upon numerous established results within the existing literature.

In this paper, we gave some generalized Young type inequalities due to Zuo and Li [J. Math. Inequal., 16 (2022), 1169-1178], and we also presented a new Young type inequality. As applications, we obtained some operator inequalities and matrix versions inequalities including the Hilbert-Schmidt norm and trace.

In this paper, we gave some generalized Young type inequalities due to Zuo and Li [J. Math. Inequal., 16 (2022), 1169-1178], and we also presented a new Young type inequality. As applications, we obtained some operator inequalities and matrix versions inequalities including the Hilbert-Schmidt norm and trace. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some improvements and generalizations of Young's inequalities on the arithmetic, geometric, and harmonic mean. For example, In addition, we obtain some new results for Young's inequality for operators. [ABSTRACT FROM AUTHOR]

Recently, some Young-type inequalities have been promoted. The purpose of this article is to give further refinements and reverses to them with Kantorovich constants. Simultaneously, according to the scalar result, we have obtained some corresponding operator inequalities and matrix versions, including Hilbert-Schmidt norm, unitary invariant norm, and trace norm can be regarded as Scalar inequality.

In this paper, we propose new refinements and reverses of real power form for Youngtype inequalities, which generalizes the recent inspired results by D. Q. Huy et al. [Linear Algebra Appl. 656 (2023), 368-384], and by Y. Ren et al. [J. Inequal. Appl. 2020 (2020), Paper No. 98, 13 p.]. Furthermore, the above refinements and reverses are continued to improve via the famous constants consisting of Kantorovich constant and Specht ratio. As applications, we establish operator versions, inequalities for unitarily invariant norms and inequalities for determinants of matrices. [ABSTRACT FROM AUTHOR]

Abstract Loewner partial order plays a very important role in metric topology and operator inequality on the open convex cone of positive invertible operators. In this paper, we consider a family G = { G n } n ∈ N of the ordered means for positive invertible operators equipped with homogeneity and properties related to the Loewner partial order such as the monotonicity, joint concavity, and arithmetic-G-harmonic weighted mean inequalities. Similar to the resolvent average, we construct a parameterized ordered mean and compare two types of mixtures of parameterized ordered means in terms of the Loewner order. We also show a relation between two families of parameterized ordered means associated with the power mean monotonic interpolating given two parameterized ordered means.

In this paper, we obtain some improvements and generalizations of Young's inequalities as the following:... for m ∈ N+ and 0 < v ≤ τ < 1. In addition, we obtain new result of Young's inequality by using the expansions of the functions (1-v)+vx-xv with 0 < x < 2. [ABSTRACT FROM AUTHOR]

Let Φ be a unital positive linear map and let A be a positive invertible operator. We prove that there exist partial isometries U and V such that | Φ (f (A)) Φ (A) Φ (g (A)) | ≤ U ∗ Φ (f (A) A g (A)) U and Φ f (A) − r Φ (A) r Φ g (A) − r ≤ V ∗ Φ f (A) − r A r g (A) − r V hold under some mild operator convex conditions and some positive numbers r. Further, we show that if f 2 is operator concave, then | Φ (f (A)) Φ (A) | ≤ Φ (A f (A)). In addition, we give some counterparts to the asymmetric Choi–Davis inequality and asymmetric Kadison inequality. Our results extend some inequalities due to Bourin–Ricard and Furuta. [ABSTRACT FROM AUTHOR]

Loewner partial order plays a very important role in metric topology and operator inequality on the open convex cone of positive invertible operators. In this paper, we consider a family G = { G n } n ∈ N of the ordered means for positive invertible operators equipped with homogeneity and properties related to the Loewner partial order such as the monotonicity, joint concavity, and arithmetic-G-harmonic weighted mean inequalities. Similar to the resolvent average, we construct a parameterized ordered mean and compare two types of mixtures of parameterized ordered means in terms of the Loewner order. We also show a relation between two families of parameterized ordered means associated with the power mean monotonic interpolating given two parameterized ordered means. [ABSTRACT FROM AUTHOR]

In this paper, we study some improvements of Young type inequalities. We obtain some reverse improvements of Young inequality. Among other results, we show a refined Young inequality. For v ∈ [0, 1] and a, b > 0, then a∇vb ≤ MvR (h)a♯vb, where Mv(t) = 1+v(1--v)... and h = b/a. And we also obtain a new Young type inequality. Furthermore, corresponding operator inequalities are also established. [ABSTRACT FROM AUTHOR]

The theory of inequality in mathematics is an extensive and interesting area of research. The classical Young inequality has a great importance in several fields of mathematics, statistics and other disciplines. Many results were obtained on refinements, generalizations and converses of the Young inequality. In this note, we mainly investigate the improved Young and its reverse inequalities. First, we present a new version of a generalization of reverse Young inequality. In addition, we also get the Young and its reverse inequalities for scalars with the classical Kantorovich constant. Finally, we establish the Young and its reverse inequalities for operators on infinite and finite dimensional Hilbert spaces. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we shall present some reverse arithmetic-geometric mean operator inequalities for unital positive linear maps. These inequalities improve some corresponding results due to Xue (J. Inequal. Appl. 2017:283, 2017).

In this paper, we show an analogue inequality of geometric concavity property for operator monotone functions involving Kantorovich constant. This inequality provides some unitarily norm inequalities involving operator monotone and operator convex functions. [ABSTRACT FROM AUTHOR]

Abstract This note aims to generalize the reverse weighted arithmetic–geometric mean inequality of n positive invertible operators due to Lawson and Lim. In addition, we make comparisons between the weighted Karcher mean and Lawson–Lim geometric mean for higher powers.

In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm |||⋅||| and A, B, X ∈ Mn(ℂ): | | | A ν X B 1 − ν + A 1 − ν X B ν 2 | | | ≤ (4 r 0 − 1) | | | A 1 2 X B 1 2 | | | + 2 (1 − 2 r 0) | | | (1 − α) A 1 2 X B 1 2 + α ( A X + X B 2) | | | , $$\begin{array}{} \begin{split} \displaystyle \Big|\Big|\Big|\frac{A^\nu XB^{1-\nu}+A^{1-\nu}XB^\nu}{2}\Big|\Big|\Big| \leq &(4r_0-1)|||A^{\frac{1}{2}}XB^{\frac{1}{2}}||| \\ &+2(1-2r_0)\Big|\Big|\Big|(1-\alpha)A^{\frac{1}{2}}XB^{\frac{1}{2}} +\alpha\Big(\frac{AX+XB}{2}\Big)\Big|\Big|\Big|, \end{split} \end{array}$$ where 1 4 ≤ ν ≤ 3 4 , α ∈ [ 1 2 , ∞) $\begin{array}{} \displaystyle \frac{1}{4}\leq \nu \leq \frac{3}{4}, \alpha \in [\frac{1}{2},\infty) \end{array}$ and r0 = min{ν, 1 – ν}. [ABSTRACT FROM AUTHOR]

In this paper, we shall present some reverse arithmetic-geometric mean operator inequalities for unital positive linear maps. These inequalities improve some corresponding results due to Xue (J. Inequal. Appl. 2017:283, 2017). [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to give a survey of the progress, advantages and limitations of various operator inequalities involving improved Young’s and its reverse inequalities related to the Kittaneh-Manasrah inequality. We also present our new progress to the related research topics. New scalar versions of Young’s inequalities are promoted, the operator version and the Hilbert-Schmidt form also get a promotion.

We generalize some inequalities for positive unital linear map as follows: Let A, B be positive operators on a Hilbert space with 0 < m ≤ A ≤ m' < M' ≤ B ≤ M. Then for every positive unital linear map Φ, μ ∊ [0, 1] and p > 0, Φp(A∇μB + 2rMm(A-1∇B-1 - A-1#B-1)) ≤ αpΦp(A#μB) and Φp(A∇μB + 2rMm(A-1∇B-1 - A-1#B-1)) ≤ αp(Φ(A)#μΦ(B))p where r = min {μ 1 - μ}, h' = M'/m' and α = max{ (M+m)2/4MmK(√h';2)R, (M+m)2/42/p MmK(√ h',2)R } . Furthermore, we give a squaring reversed Karcher mean inequality involving positive linear map. [ABSTRACT FROM AUTHOR]

We obtain a refined Young type inequality in this paper. The conclusion is presented as follows: Let A, B ∈ B(𝓗) be two positive operators and p ∈ [0, 1], then A ♯ p B + G ∗ (A ♯ p B) G ≤ A ∇ p B − 2 r (A ∇ B − A ♯ B) , $$\begin{array}{} \displaystyle A\sharp_p B+G^*(A\sharp_p B)G\le A\nabla_p B-2r(A\nabla B-A\sharp B), \end{array}$$ where r = min{p, 1 – p}, G = L (2 p) 2 $\begin{array}{} \displaystyle \frac{\sqrt{L(2p)}}{2} \end{array}$ A–1S(A|B), L(t) is periodic with period one and L(t) = t 2 2 1 − t t 2 t $\begin{array}{} \displaystyle \frac{t^2}{2}\left(\frac{1-t}{t} \right)^{2t} \end{array}$ for t ∈ [0, 1]. Moreover, we give the s-th powering of two inequalities related to the above one with s > 0 which refines Lin's work. In the mean time, we present an inequality involving Hilbert-Schmidt norm. [ABSTRACT FROM AUTHOR]

In this short note, we give the refined Young inequality with Specht's ratio by only elementary and direct calculations. The obtained inequality is better than one previously shown by the author in 2012. In addition, we give a new property of Specht's ratio. These imply an alternative proof of the refined Young ineqaulity shown by author in 2012. We also give a remark on the relation to Kantorovich constant. Finally, we give a proposition for a corresponding general function. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we employ iteration on operator version of the famous Young inequality and obtain more arithmetic-geometric mean inequalities and the reverse versions for positive operators. Concretely, we obtain refined Young inequalities with the Kantorovich constant, the reverse ratio type and difference type inequalities for arithmetic-geometric operator mean under different conditions.

In present paper, we give some new reverses of the Young type inequalities which were established by X. Hu and J. Xue [7] via Kantorovich constant. Then we apply these inequalities to establish corresponding inequalities for the Hilbert-Schmidt norm and the trace norm. [ABSTRACT FROM AUTHOR]

The main objective of the present paper, is to obtain some new versions of Young-type inequalities with respect to two weighted arithmetic and geometric means and their reverses, using two inequalities K (b a , 2) r ≤ a ∇ ν b a ♯ ν b ≤ K (b a , 2) R , $$\begin{array}{} \displaystyle K\Big(\frac{b}{a},2\Big)^r\leq\frac{a\nabla_{\nu}b}{a\sharp_{\nu}b}\leq K\Big(\frac{b}{a},2\Big)^R, \end{array}$$ where r = min{ν, 1 – ν}, R = max{ν,1 – ν} and K(t,2) = (t + 1) 2 4 t $\begin{array}{} \displaystyle \frac{(t+1)^2}{4t} \end{array}$ is the Kantorovich constant, and e (h − 1 , ν) ≤ a ∇ ν b a ♯ ν b ≤ e (h , ν) , $$\begin{array}{} \displaystyle e(h^{-1},\nu)\leq\frac{a\nabla_{\nu}b}{a\sharp_{\nu}b}\leq e(h,\nu), \end{array}$$ where h = max { a b , b a } $\begin{array}{} \displaystyle \{\frac{a}{b},\frac{b}{a}\} \end{array}$ and e(t,ν) = exp (4ν(1 – ν)(K(t,2)–1) (1 − 1 2 t)). $\begin{array}{} \displaystyle (1-\frac{1}{2t})\big). \end{array}$ Also some operator versions of these inequalities and some inequalities related to Heinz mean are proved. [ABSTRACT FROM AUTHOR]

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 purpose of this paper is to give a survey of the progress, advantages and limitations of various operator inequalities involving improved Young's and its reverse inequalities related to the Kittaneh-Manasrah inequality. We also present our new progress to the related research topics. New scalar versions of Young's inequalities are promoted, the operator version and the Hilbert-Schmidt form also get a promotion. [ABSTRACT FROM AUTHOR]

In this paper, we present two inequalities of matrix norms. The first one is a generalization of the inequality shown in [J Math Inequal 10 (2016) 1119-1122], and the second one is a refinement of an inequality obtained by Zou [Numer Math J Chinese Univ 38 (2016) 343-349]. [ABSTRACT FROM AUTHOR]

In this paper, we employ iteration on operator version of the famous Young inequality and obtain more arithmetic-geometric mean inequalities and the reverse versions for positive operators. Concretely, we obtain refined Young inequalities with the Kantorovich constant, the reverse ratio type and difference type inequalities for arithmetic-geometric operator mean under different conditions. [ABSTRACT FROM AUTHOR]

We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean. [ABSTRACT FROM PUBLISHER]

In this paper, from the viewpoint of the Ando–Hiai inequality, we make a comparison among three geometric means: The Ando–Li–Mathias geometric mean, the Karcher mean and the chaotic geometric mean ofpositive definite matrices. Among others, we show complements of the-variable Ando–Hiai inequality for the Ando–Li–Mathias geometric mean by means of the Kantorovich constant. [ABSTRACT FROM AUTHOR]

In this paper we consider Jensen's operator, which includes bounded self-adjoint operator on Hilbert space, and investigate its properties. Due to derived properties, we find lower and upper bound for Jensen's operator, which are multiples of non-weighted Jensen's operator. On the other hand, we also establish some bounds for spectra of Jensen's operator by means of discrete Jensen's functional. The obtained results are then applied to operator means. In such a way, we get refinements and conversions of numerous mean inequalities for Hilbert space operators. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we present an alternative proof of the power convergence of the symmetrization procedure on the weighted geometric mean due to Lawson and Lim in [J. Lawson and Y. Lim, A general framework for extending means to higher orders, preprint] by using a limiting process due to Ando-Li-Mathias in [T. Ando, C.-K. Li, R. Mathias, Geometric means, Linear Algebra Appl. 385 (2004) 305–334]. As applications, we obtain a reverse of the weighted arithmetic-geometric mean inequality of n-operators via Kantorovich constant: For any positive integer , let be positive invertible operators on a Hilbert space H such that for some scalars . Then [Copyright &y& Elsevier]