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        Questions tagged [inequalities]

        for questions involving inequalities.

        3
        votes
        1answer
        172 views

        A certain generalisation of the golden ratio

        Consider a real number $a \ge 1$, and let $g(a)$ be the unique positive solution $x>0$ of $x^a - x^{a-1} - 1 = 0.$ We have $g(1) = 2$, $g(2) = {1+\sqrt{5}\over2}$ (the golden ratio), and $g$ is ...
        2
        votes
        1answer
        113 views

        Construct examples satisfying some inequalities [on hold]

        How do I construct two vectors $a,b\in \mathbb{R}^{n}$, $a=(a_1,a_2,\ldots, a_n)^T$ and $b=(b_1,b_2,\ldots, b_n)^T$ which satisfy in the following conditions ?\begin{align} & a_ib_i\geq 1,a_ib_j&...
        12
        votes
        3answers
        299 views

        (Sharp) inequality for Beta function

        I am trying to prove the following inequality concerning the Beta Function: $$ \alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0, $$ where as usual $B(a,b) = \...
        4
        votes
        3answers
        252 views

        Question about an inequality described by matrices

        Let $A=(a_{ij})_{1 \le i, j \le n}$ be a matrix such that $\sum_\limits{i=1}^{n} a_{ij}=1$ for every $j$, and $\sum_\limits{j=1}^n a_{ij} = 1$ for every $i$, and $a_{ij} \ge 0$. Let $$\begin{equation} ...
        8
        votes
        0answers
        207 views

        How localized can a polynomial be in the L1 norm?

        Let $0<s<2$ be a parameter, $\Omega = [-1,1]$, and $\Omega_s\subset \Omega$ be a set of measure $s$. I would like to bound the following ratio from above: $$\sup_{p\in\mathcal{P}_n} \frac{\...
        15
        votes
        0answers
        215 views

        An inequality for matrix norms

        Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically: Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
        1
        vote
        0answers
        47 views

        Can the Sobolev Inequalities be derived from the Weighted Hardy Inequality, or vice versa?

        Here, for the weighted version of the Hardy Inequality, I refer to Muckenhoupt's formulation in Theorem 1 of 1 Sobolev Inequality: $$C_d \int_{\mathbb{R}^d} \vert \nabla \phi \vert^2 \geq \left( \...
        1
        vote
        1answer
        48 views

        Randomly scaled random variables

        Consider two possibly correlated scalar random variables $N$ and $X$. It is known that $1\leq N \leq N_{\max}$. Given that $\mathbb{E}[NX]\leq 0$, does it always hold that $\mathbb{E}[X] \leq 0$? ...
        5
        votes
        1answer
        123 views

        Bounds on the L^1 norm of a discrete Fourier spectrum

        I am dealing with a function $f$ of the form \begin{equation} f(t):=\sum_{k=1}^Na_ke^{\mathrm{i}\phi_k t} \end{equation} and I have a promise that \begin{equation} 0\leq f(t)\leq C\;\;\;\text{for all}...
        0
        votes
        0answers
        55 views

        Energy estimates involving test functions for weak solutions of PDE problems

        I was reading an article on Arxiv.org about Navier-Stokes system ([Breit]) and I stumble on this sentence on the second page: "A weak (in the PDE sense) solution satisfying the energy inequality ...
        5
        votes
        1answer
        142 views

        Сoincidence of discrete random variables

        Let $\xi, \eta$ be a discrete random values and $\mathbb E| ξ |$, $\mathbb E | η | < +\infty$, and any value of these values ??are accepted with a non-zero probability. How to prove that from $\...
        2
        votes
        1answer
        93 views

        The blow-up rate of a nonlinear oscillator

        (Related to this Math.SE question.) For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$ ...
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        votes
        0answers
        121 views

        Proving that the triangle inequality holds for a metric on $\mathbb{C}$ [closed]

        Following problem had post mathstack three years ago,and until now no one solve it.so I ask here,Thanks for you help. let $$f(x,y)=\dfrac{|x-y|}{\sqrt{|x|^2+1}+\sqrt{|y|^2+1}}$$ show that $$f(x,y)+f(...
        3
        votes
        1answer
        83 views

        Hoeffding's inequality for Hilbert space valued random elements

        Suppose that $\mathbb H$ is a separable Hilbert space and $X_1,\ldots,X_n$ are independent zero mean $\mathbb H$-valued random elements such that $\|X_i\|\le s$ for each $1\le i\le n$, where $\|\cdot\|...
        2
        votes
        1answer
        197 views

        Complicated bound after using Stirling's approximation

        I have this inequality $$\frac{1}{a}\exp\bigl\{-\frac{4}{h^2}\bigr\} \geq \frac{1}{f}$$ where $$ a \leq \Bigl(\pi^{d/2}\Gamma(\frac{1}{2}d+1)^{-1} + 1\Bigr) \left(\frac{h^{d+1}}{2} \Gamma \left(\frac{...

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