Related papers: Sharp Inequalities between Harmonic, Seiffert, Qua…

We prove amongs others results that the harmonic mean of $\Gamma_q(x)$ and $\Gamma_q(1/x)$ is greater than or equal to $1$ for arbitrary $x > 0$ and $q\in J$ where $J$ is a subset of $[0,+\infty)$. Also, we prove that for there is…

For $p,q\geq2$, the Hardy and Littlewood inequalities for real bilinear forms, in its unified formulation, assert that there is a constant $C_{p,q}\geq1$ such that \begin{equation}…

Recently, Yanyan Li and Xukai Yan showed the following interesting Hardy inequalities with anisotropic weights: Let $n\geq 2$, $p \geq 1$, $p\alpha > 1-n$, $p(\alpha + \beta)> -n$, then there exists $C > 0$ such that…

Let $A,B\in \mathbb{B}(\mathscr{H})$ be such that $0<b_{1}I \leq A \leq a_{1}I$ and $0<b_{2}I \leq B \leq a_{2}I$ for some scalars $0<b_{i}< a_{i},\;\; i=1,2$ and $\Phi:\mathbb{B}(\mathscr{H})\rightarrow\mathbb{B}(\mathscr{K})$ be a…

Let $\mathcal{H} \subset \mathcal{H}_{n,d} := \mathbb{R}[x_1$,$\ldots$, $x_n]_d$ be a vector space, and $A$ be a compact semialgebraic subset of $\mathbb{P}_{\mathbb{R}}^{n-1}$. We shall study some PSD cones $\mathcal{P} = \mathcal{P}(A$,…

The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x=(x_1,...,x_g). The Laplacian Lap[p,h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have…

For fixed $s\geq 1$ and $t_{1},t_{2}\in(0,1/2)$ we prove that the inequalities $G^{s}(t_{1}a+(1-t_{1})b,t_{1}b+(1-t_{1})a)A^{1-s}(a,b)>AG(a,b)$ and $G^{s}(t_{2}a+(1-t_{2})b,t_{2}b+(1-t_{2})a)A^{1-s}(a,b)>L(a,b)$ hold for all $a,b>0$ with…

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $\alpha > 1$, $\beta > 1$ with $\alpha + \beta = 2^*(s)$, and $\kappa \in \mathbb{R}$. For any…

For the observed t --> W b decay, an intensity-ratio equivalence-theorem for two Lorentz-invariant couplings is shown to be related to symmetries of tWb-transformations. Three explicit tWb-transformations, A_{+}=M A_{SM}, ... relate the…

In this paper, we study the following biharmonic equations:% $$ \left\{\aligned&\Delta^2u-a_0\Delta u+(\lambda b(x)+b_0)u=f(u)&\text{ in }\bbr^N,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{\lambda})}% $$ where $N\geq3$,…

In the paper, by finding linear relations of differences between some means, the authors supply a unified proof of some double inequalities for bounding Neuman-S\'andor means in terms of the arithmetic, harmonic, and contra-harmonic means…

We are concerned with the study of the twin non-local inequalities featuring non-homogeneous differential operators $$\displaystyle -\Delta^2 u + \lambda\Delta u \geq (K_{\alpha, \beta} * u^p)u^q \quad\text{ in } \mathbb{R}^N (N\geq 1),$$…

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy's inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^2\,dx\leq d(a,b)\,\int_a^b [f(x)]^2 dx $$ and $$…

Using the properties of geometric mean, we shall show for any $0\le \alpha ,\beta \le 1$, \[f\left( A{{\nabla }_{\alpha }}B \right)\le f\left( \left( A{{\nabla }_{\alpha }}B \right){{\nabla }_{\beta }}A \right){{\sharp}_{\alpha }}f\left(…

We prove that for $p\in (0,1]$, the double inequality% \begin{equation*} \tfrac{1}{3p^{2}}\cos px+1-\tfrac{1}{3p^{2}}<\frac{\sin x}{x}<\tfrac{1}{% 3q^{2}}\cos qx+1-\tfrac{1}{3q^{2}} \end{equation*}% holds for $x\in (0,\pi /2)$ if and only…

Assume that $p\in(1,\infty]$ and $u=P_{h}[\phi]$, where $\phi\in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^{n})$. Then for any $x\in \mathbb{B}^{n}$, we obtain the sharp inequalities $$ |u(x)|\leq…

In this note, we derive non trivial sharp bounds related to the weighted harmonic-geometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of…

Suppose that $c,d,\alpha,\beta$ are real numbers satisfying the inequalities $1<d<c<79/71$ and $1<\alpha<\beta<6^{1-d/c}$. In this paper, it is proved that, for sufficiently large real numbers $N_1$ and $N_2$ subject to $\alpha\leqslant…

Using recent results concerning the homogenization and the Hardy property of weighted means, we establish sharp Hardy constants for concave and monotone weighted quasideviation means and for a few particular subclasses of this broad family.…

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…