Related papers: Extensions of interpolation between the arithmetic…

Consider the following noncommutative arithmetic-geometric mean inequality: given positive-semidefinite matrices $\mathbf{A}_1, \dots, \mathbf{A}_n$, the following holds for each integer $m \leq n$: $$ \frac{1}{n^m}\sum_{j_1, j_2, \dots,…

In this paper, for $0<\alpha<1$, $p>0$ and positive semidefinite matrices $A,B\ge0$, we consider the quasi-extension $\mathcal{A}_{\alpha,p}(A,B):=((1-\alpha)A^p+\alpha B^p)^{1/p}$ of the $\alpha$-weighted arithmetic matrix mean, and the…

Some inequalities for positive linear maps on matrix algebras are given, especially asymmetric extensions of Kadison's inequality and several operator versions of Chebyshev's inequality. We also discuss well-known results around the matrix…

Given $E_0, E_1, F_0, F_1, E$ rearrangement invariant function spaces, $a_0$, $a_1$, $b_0$, $b_1$, $b$ slowly varying functions and $0< \theta_0<\theta_1<1$, we characterize the interpolation spaces $$(\overline{X}^{\mathcal…

This paper undertakes a thorough investigation of matrix means interpolation and comparison. We expand the parameter $\vartheta$ beyond the closed interval $[0,1]$ to cover the entire positive real line, denoted as $\mathbb{R}^+$.…

For a $m\times n$ matrix $B=(b_{ij})_{m\times n}$ with nonnegative entries $b_{ij}$ and any $k\times l-$submatrix $B_{ij}$ of $B$, let $a_{B_{ij}}$ and $g_{B_{ij}}$ denote the arithmetic mean and geometric mean of elements of $B_{ij}$…

In this paper, we present some extensions of the Young and Heinz inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with matrices. More precisely, for two…

We prove an inequality for unitarily invariant norms that interpolates between the Arithmetic-Geometric Mean inequality and the Cauchy-Schwarz inequality.

For $n$ positive numbers ($a_k$, $1\leq k \leq n$), enhanced inequalities about the arithmetic mean ($A_n \equiv \frac{\sum_ka_k}{n}$) and the geometric mean ($G_n\equiv \sqrt[n]{\Pi_ka_k}$) are found if some numbers are known, namely,…

There are various generalizations of the geometric mean $(a,b)\mapsto a^{1/2}b^{1/2}$ for $a,b\in \mathbb{R}^+$ to positive matrices, and we consider the standard positive geometric mean $(X,Y)\mapsto…

This note proves the following inequality: if $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $A_1,A_2,\cdots,A_n$, \begin{equation} \frac{1}{n^3}\Big\|\sum_{j_1,j_2,j_3=1}^{n}A_{j_1}A_{j_2}A_{j_3}\Big\|…

In this note we present a refinement of the AM-GM inequality, and then we estimate in a special case the typical size of the improvement.

In this paper, we use some standard numerical techniques to approximate the hypergeometric function $$ {}_2F_1[a,b;c;x]=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^2}{2!}+\cdots $$ for a range of parameter triples $(a,b,c)$ on the…

We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \in \{1\} \cup [2, \infty)$, positive semi-definite matrices $A_i,\…

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…

In this paper we develop a general method for improving Jensen-type inequalities for convex and, even more generally, for piecewise convex functions. Our main result relies on the linear interpolation of a convex function. As a consequence,…

Let $A, B$ and $X$ be $n\times n$ matrices such that $A, B$ are positive semidefinite. We present some refinements of the matrix Cauchy-Schwarz inequality by using some integration techniques and various refinements of the Hermite--Hadamard…

Let $A_i$ and $B_i$ be positive definite matrices for all $i=1,\cdots,m.$ It is shown that $$\left|\left|\sum_{i=1}^m(A_i^2\sharp…

Let $m_{2}<m_{1}$ be two given nonnegative integers with $n=m_{1}+m_{2}+1$. For suitably differentiable $f$, we let $P,Q\in \pi_{n}$ be the Hermite polynomial interpolants to $f$ which satisfy $P^{(j)}(a)=f^{(j)}(a),j=0,1,...,m_{1}$ and…

We shall give a refinement of the arithmetic-geometric mean inequality.