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        Questions tagged [inequalities]

        for questions involving inequalities.

        5
        votes
        0answers
        35 views

        On the relation between elementary symmetric polynomial inequalities and $p$-norms

        Let $x, y \in \mathbb{R}^n_+$, and let $e_k$ denote the $k$th elementary symmetric polynomial. It is known (see e.g. [1]) that if \begin{align} e_1(x) &= e_1(y),\tag{1}\\ e_k (x) &\leq e_k (y)...
        6
        votes
        1answer
        174 views

        How to prove this inequality of Karamata type?

        Question 1: Let $x_{i}>0$, ($i=1,2,\cdots,n$) and such that $$x_{1}+x_{2}+\cdots+x_{n}=\pi.$$ Show that $$ \dfrac{\sin{x_{1}}\sin{x_{2}}\cdots\sin{x_{n}}}{\sin{(x_{1}+x_{2})}\sin{(x_{2}+x_{3})}\...
        4
        votes
        1answer
        112 views

        About one integral inequality

        Set $\phi (x) = u(x)+iv(x)$, $x=(x_1,\ldots,x_N)$, a $T$-periodic function in $H^1_\text{loc}(\mathbb{R}^N)$, that is $\phi (x) = \phi (x_1 + T ,\ldots, X_N + T)$ for all $x$ and where $u = \...
        4
        votes
        1answer
        75 views

        Lipschitz property of matrix function only depending on singular values

        Let $f$ be a function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ such that there exists another symmetric function $g$ (invariant under permutation of coordinates) from $\mathbb{R}^{n}$ to $\mathbb{...
        1
        vote
        0answers
        52 views

        A comprehensive list of random walk inequalities?

        I am interested in finding a comprehensive list of all noticeable random walk inequalities. ie. $S_n = \sum_{k\leq n} X_i$ for i.i.d symmetric $X_i$ I can only seem to find books/papers that list ...
        1
        vote
        0answers
        49 views

        Tail decay of the norm of spherical distribution

        Let the distribution of the random vector $X\in\mathbb{R}^n$ be orthogonally invariant. I'm considering the following expectation for any vector $\mu\in\mathbb{R}^n$: $$f(\|\mu\|)=\mathbb{E}\left[\exp(...
        -2
        votes
        0answers
        50 views

        How can I show $1+c_n^2-a_n^2-b_n^2\geq 0$? [on hold]

        $$c_n=1-2t_n+2t_{n-1}-...\pm2t_1~,~b_n=1-2t_j+2t_{j-1}-...~,~~~~~~$$ $$a_n=1-2t_i+2t_{i-1}-...$$ $0<t_1<t_2<t_3....<t_n<1$ and $j\in J$ and $i\in I$ and $I\cup J=\{1,2,3,...,n\}$ and $I\...
        1
        vote
        1answer
        99 views

        Moment generating function of random unit vector

        Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of $$\mathbb{E}[\exp(X^Tv)]$$ for any $v$?
        2
        votes
        0answers
        66 views

        Gagliardo-Nirenberg inequality for periodic functions?

        I am interested in Gagliardo-Nirenberg type inequality (see https://en.m.wikipedia.org/wiki/Gagliardo%E2%80%93Nirenberg_interpolation_inequality) for functions in the space $$H^1_T(\mathbb{R}^n)=\...
        0
        votes
        0answers
        51 views

        Ratio of exponentially weighted Selberg integrals

        I'm interested in bounding the following ratio of integral: $$\frac{\int_{0<x_k<...<x_1<1}\prod_{i=1}^kx_i^{m-\frac{k+1}{2}}\prod_{i<j}(x_i-x_j)\exp(-\sum_{i=1}^kw_ix_i)}{\int_{0<x_k&...
        4
        votes
        1answer
        143 views

        Relation between two different functionals: $\lVert p^{-\max}_{-\varepsilon}\rVert$ and $\kappa_{p}^{-1}(1-\varepsilon)$

        Given a non-negative sequence $p=(p_i)_{i\in\mathbb{N}}\in \ell_1$ such that $\lVert p\rVert_1 = 1$,we define the two following quantities, for every $\varepsilon \in (0,1]$. Assuming, without loss ...
        0
        votes
        1answer
        81 views

        a simple functional inequation

        Is there some general solution to the functional inequality: f(x*y) <= y*f(x) + x*f(y) Where x and y are in [0,1]? I can find many particular solutions ...
        3
        votes
        1answer
        115 views

        When does the spectral radius strictly increase?

        For bounded linear operators $A$ and $B$ on a Banach space $X$, I'm looking for results which imply that $r(A) < r(A+B)$ (note the strict inequality), where $r(A)$ denotes the spectral radius of $A$...
        2
        votes
        0answers
        58 views

        Is there a software to solve functional inequalities?

        Suppose I have some inequalities that my function (say $\mathbb R\to \mathbb R$) needs to satisfy, like $\forall x,y\; f(x)+f(y)\le f(x+y)$ and $f(1)=0$. Is there some software that can find solutions/...
        1
        vote
        1answer
        140 views

        Inequalities for moments of a certain integral

        Let $X(t)$ be a stationary Gaussian process, $EX(t)=0$, the correlation function $R(\tau)$ is given. What bounds from above can be given for the $p$-th moment ($p>0, p \in \mathbb{R}$) of the ...

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