Related papers: Matrix Inequalities between $f(A)\sigma f(B)$ and …

We give variations on Ando's result comparing $f(B)-f(A)$ and $f(|B-A|)$ with respect to unitarily invariant norms on matrices.

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \| f(z) \|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty…

Let $A, B$ be positive definite matrices, $p=1, 2$ and $r\ge 0$. It is shown that \begin{equation*} ||A+ B + r(A\sharp_t B+A\sharp_{1-t} B)||_p \le ||A+ B + r(A^{t}B^{1-t} + A^{1-t}B^t)||_p. \end{equation*} We also prove that for positive…

In recent literature concerning integer partitions one can find many results related to both the Bessenrodt-Ono type inequalities and log-concavity property. In this note we offer some general approach to this type of problems. More…

Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $ f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $f\in \mathcal{S}$, In 1999, Ma proposed the…

Let $f$ be analytic in the unit disk and $\mathcal{S}$ be the subclass of normalized univalent functions with $f(0) = 0$, and $f'(0) = 1$. Let $F$ be the inverse function of $f$, given by $F(w)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some…

Let $F(x) := (f_{ij}(x))_{i,j=1,\ldots,p},$ be a real symmetric polynomial matrix of order $p$ and let $f(x)$ be the largest eigenvalue function of the matrix $F(x).$ We denote by ${\partial}^\circ f(x)$ the Clarke subdifferential of $f$ at…

Let $\BS_1,...,\BS_n$ be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter $p\in(0,1)$. Let $m_*(p):=(1+p+2p^2)/(2\sqrt{p-p^2}+4p^2)$ if $0<p\le 1/2$ and $m_*(p):=1$ if…

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…

We mainly establish a monotonicity property between some special Riemann sums of a convex function $f$ on $[a,b]$, which in particular yields that $\frac{b-a}{n+1}\sum_{i=0}^n f\left(a+i\frac{b-a}{n}\right)$ is decreasing while…

Assuming a unitarily invariant norm $|||\cdot|||$ is given on a two-sided ideal of bounded linear operators acting on a separable Hilbert space, it induces some unitarily invariant norms $|||\cdot|||$ on matrix algebras $\mathcal{M}_n$ for…

Let $A$ be a positive semidefinite $m\times m$ block matrix with each block $n$-square, then the following determinantal inequality for partial traces holds \[ (\mathrm{tr} A)^{mn} - \det(\mathrm{tr}_2 A)^n \ge \bigl| \det A -…

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…

Let $\mathcal{A}$ be the family of analytic and normalized functions in the open unit disc $|z|<1$. In this article we consider the following classes \begin{equation*} \mathcal{R}(\alpha,\beta):=\left\{ f\in \mathcal{A}: {\rm…

The M\"obius invariant space $\mathcal{Q}_p$, $0<p<\infty$, consists of functions $f$ which are analytic in the open unit disk $\mathbb{D}$ with $$ \|f\|_{\mathcal{Q}_p}=|f(0)|+\sup_{w\in \D} \left(\int_\D |f'(z)|^2(1-|\sigma_w(z)|^2)^p…

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 $f \geq 0$ be operator monotone on $[0, \infty)$. In this paper we prove that for any unitarily-invariant norm $|||-|||$ on $M_n(\mathbb{C})$ and matrices $A, B, X \in M_n(\mathbb{C})$ with $A, B \geq 0$ and $|||X||| \leq 1$,…

For any positive invertible matrix $A$ and any normal matrix $B$ in $M_{n}({\Bbb C})$, we investigate whether the inequality $ ||A\sharp (B^{*}A^{-1}B)||\geq ||B|| $ is true or not, where $\sharp$ denotes the geometric mean and $||\cdot||$…

For $n$-by-$n$ and $m$-by-$m$ complex matrices $A$ and $B$, it is known that the inequality $w(A\otimes B)\le\|A\|w(B)$ holds, where $w(\cdot)$ and $\|\cdot\|$ denote, respectively, the numerical radius and the operator norm of a matrix. In…

Let $f\colon\mathbb{N}\rightarrow\mathbb{C}$ be an arithmetic function and consider the Beatty set $\mathcal{B}(\alpha) = \lbrace\, \lfloor n\alpha \rfloor : n\in\mathbb{N} \,\rbrace$ associated to a real number $\alpha$, where…