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      Questions tagged [inequalities]

      for questions involving inequalities.

      3
      votes
      3answers
      140 views

      A challenging inequality that involves the digamma function and polygamma functions

      Let $f(x)=x \psi(x+1)$, where $\psi$ is the digamma function. Define $$g(x)=(f(ax)+f((1-a)x)-f(x))-(f(ax+by)+f((1-a)x+(1-b)y)-f(x+y)),$$ where $0\le a,b\le 1$ and $x,y\ge 0$. How to show that $g(x)$ ...
      1
      vote
      0answers
      123 views

      Lower bound for the $L^2$ norm of a polynomial / hypergeometric function

      Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
      1
      vote
      1answer
      55 views

      Hanson-Wright inequality (quadratic form concentration inequality) for bounded random vectors

      Is there a concentration inequality for quadratic forms of bounded random vectors $X \in [-1, 1]^n$ with zero mean and given covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ but otherwise ...
      8
      votes
      1answer
      561 views

      A curious inequality concerning binomial coefficients

      Has anyone seen an inequality of this form before? It seems to be true (based on extensive testing), but I am not able to prove it. Let $a_1,a_2,\ldots,a_k$ be non-negative integers such that $\sum_i ...
      0
      votes
      0answers
      57 views

      Does there exist $\alpha>0, \beta\in (0,1)$ such that $\dfrac{\sum_{k=1}^n a_k}{n}\le \alpha (a_1\cdots a_n)^{1/n} + \beta \max_i(a_i)$ holds?

      Let $a_1\ge a_2\ge \cdots\ge a_n\ge 0$ be given non-negative numbers. My question is the following: Is there any $\beta \in(0,1),\ \alpha>0$, such that $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\...
      1
      vote
      1answer
      225 views

      Prove that $x^{y^x}>y^{x^y}$ for $x>y>0.$ [closed]

      Let $x>y>0$. Prove that $$x^{y^x}>y^{x^y}.$$ My attempts: Let $1>x>y>0$. In this case it's enough to prove that $$y^x<x^y$$ or $$x\ln\frac{1}{y}>y\ln\frac{1}{x},$$ which is ...
      2
      votes
      1answer
      207 views

      An isoperimetric inequality for curve in the plane?

      Let $f(x,y)=0$ be a (smooth) simple closed curve $C$ on the plane and $R$ the region bounded by $C$ (appropriately oriented). Assume the origin lies in the interior of $R$. QUESTION. Let $r=\sqrt{x^...
      1
      vote
      2answers
      134 views

      Infimum upper bound

      Let $h:[0,1]\to[0,1]$ be a $\mathcal{C^1}$ function such that $h'(x)<0$ for all $x\in(0,1)$. I am trying to show that (not sure if it is true): $$ \inf_{f\in \mathcal{H}}\left(\int_0^1\int_0^1 h(|...
      0
      votes
      0answers
      78 views

      Definition for Norm of a Tensor [on hold]

      I am reading a book on differential geometry and at one point the following statement is made: $|\text{Ric} + \text{Hess}(f)|^2 \geq \frac{1}{n}|R+\Delta f|^2.$ The justification is that 'we have ...
      5
      votes
      1answer
      141 views

      Binomial Distributions and Inequality

      Suppose $c>2$ is a real number, and $x$ solves equation $f(\frac{1}{2}+x,m,2m)=(2m+1)cx$ for some $m>0$ integer number, where $f$ is probability mass function for binomial distribution, with ...
      6
      votes
      1answer
      140 views

      An inequality for rearrangement-style sums

      The following is a holdover from my math contest days that I never got around to solve. We will use the notation $\left[ k\right] $ for the set $\left\{ 1,2,\ldots,k\right\} $ whenever $k$ is a ...
      2
      votes
      2answers
      156 views

      An inequality on elementary symmetric polynomial of eigenvalues

      For $A \in \mathbb{R}^{n\times n}$, I am wondering if the following inequality holds, $$\text{tr}(A)^2 - \text{tr}(A^2) \leq ||A||_*^2 - ||A||_F^2$$ This is equivalent to the following inequality on ...
      0
      votes
      1answer
      64 views

      Inequality involving product-of-minus vs minus-of-product for positive integers

      I'm encountering this inequality for dimensionality reduction problem. The simplified form looks as follows: Consider positive integers $a_1$, $a_2$, $b_1$ and $b_2$ where $a_1>b_1$ and $a_2>...
      12
      votes
      3answers
      2k views

      Does anyone recognize this inequality?

      In some paper the authors make use of the following inequality without further explanation: Let $x\in\mathbb{R}^n$ with $x_1\le\cdots\le x_n$ and $\alpha\in[0,1]^n$ with $\sum_{i=1}^n \alpha_i=N\in\{1,...
      1
      vote
      1answer
      58 views

      Good UPPER bounds for $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i$ where $(p_i)_i$ is a probability vector

      Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector. Question $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$ Observation This paper allows us to ...

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