# Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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### Conditional Berry–Esseen Inequality

From Wikipedia, the Berry-Essen Inequality is given by $$\sup_{x\in \mathbb{R}}|\text{Pr}(X\leq z)-\Phi(z)|\leq \frac{C E(|X|^3)}{E(X^2)\sqrt{n}}.$$ From a statistics paper that I am reading, I am ...
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### Minkowski inequality and infinite sum

I really need your help, we have $u_{2k-1}(t)=\phi_{2k-1}\cos(\lambda_{k}t)+\frac{1}{\lambda_k}\psi_{2k-1}\sin(\lambda_{k}t)+\frac{1}{\lambda_k}\int_{0}^{t}F_{2k-1}(\tau;u)\sin\lambda_{k}(t-\tau)d\tau$...
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### Find the relationship among adjusted 3-pointer (not parameter, but formula like Pythagenport, Pythagenpat), goals scored and goals allowed in football

I was seeking a formula to relate goals scored and goals allowed to points in association football. In association football, three points are awarded to the team winning a match, with no points ...
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### I am trying to find an uncommonly cited paper called: "Morrey inequality for Sobolev functions with mean of zero(?)" (I am not sure about given name.)

I am trying to find an uncommonly cited paper called: "Morrey inequality for Sobolev functions with mean of zero(?)" (Or "mean is zero", not so sure about given name.) I can ...
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### If $f'(x) − f'(y) ≤ 3|x − y|$ then show that $|f(x)−f(y)-f'(y)(x-y)|≤\frac{3}{2}(x-y)^2$

Let $f : \mathbb R \rightarrow \mathbb R$ be a twice differentiable function. Suppose that for all $x, y \in \mathbb R$ the function $f$ satisfies $f'(x) − f'(y) ≤ 3|x − y|$ then show that for all $x$...
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### A problem concerning a divergent function on $[0, 1]$

This problem was posted on another forum and was given at the 1992 Miklós Schweitzer Competition. This competition is known for its very difficult problems and this one seems no exception. I also can'...
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### Is there a simple proof for $\sup_{|x|\le \delta} \frac{|e^{ax} - 1|}{|x|} \le \frac{e^{|a|\delta}}{\delta}$?

Suppose that $a \in \mathbb{R}$. There's an inequality $$\sup_{|x|\le \delta} \frac{|e^{ax} - 1|}{|x|} \le \frac{e^{|a|\delta}}{\delta},$$ see "Testing statistical hypotheses" by E. ...
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### For help an inequality that for large $n$, whether $\frac{n\choose \big[\frac{n}{2\log_2n}\big]-1}{(\sqrt{2})^n}\geq(1+\beta)^n$ for some $\beta>0$

Can anyone help me to prove that for large $n$ whether $$\dfrac{n\choose \big[\frac{n}{2\log_2n}\big]-1}{(\sqrt{2})^n}\geq(1+\beta)^n$$ for some $\beta>0$ where and $\big[x\big]$ denotes the ...
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### Intuition behind the entropy functional for curvature bounds

What is the intuition behind defining in metric measure spaces a Ricci curvature bound via convexity or concavity of the entropy functional? I know that this is made because in Riemannian manifolds ...
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Let $f(n)=n!$ and $g(n)=\binom{2n}{n}$. Obviously $f(7)\gt g(7)$ and $f(6)\lt g(6)$. The functions are increasing on $n\in (1,\infty)$, $n$ being an integer. But this is not enough to prove $$n!\gt\... -1 votes 1 answer 105 views ### Ask for help a inequality problem, does \sum\limits_{i=0}^{k-2}\log_2\left(\frac{n-i}{k-i-1}\right)>c\cdot n [closed] Can anyone help me to give me a detailed proof (or disproof) of the following \sum\limits_{i=0}^{k-2}\log_2\left(\frac{n-i}{k-i-1}\right)>c\cdot n for some constant c>0, where k=\Big[\frac{... 5 votes 1 answer 200 views ### Spectral gap and Poincaré inequality Consider the PDE$$\partial_t u = L u$$where L = \Delta + \nabla V \cdot \nabla  is a self-adjoint operator. I read that if L has a spectral gap \lambda > 0 then "[convergence of the ... 0 votes 1 answer 27 views ### Solution for a particular inequality I'm following up on this question on MathOverflow. By taking some particular functions, a(.),\,b(.) and K(.,.). This problem is equivalent to the following inequality:$$4\left \| r \right \|_{\...
Let $h:(0,1) \to (-\infty,0)$ be a $C^1$ function, with $h'>0$. I am looking for sufficient conditions on $h$ that imply the existence of a $C^1$ decreasing* function $g:(0,1) \to (0,1)$ with \$g'&...