Questions tagged [inequalities]
for questions involving inequalities, upper and lower bounds.
1,877 questions
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Inequality involving powers of sums over 2-subsets
The following arose in a project about design enumeration.
Let $V=\{1,2,\ldots,v\}$. For any $r$, $\binom Vr$ denotes set of all $r$-subsets of $V$.
There is a real number $\theta_e$ for each $e\in\...
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Poincaré inequality on domains bounded in one direction via compactness
Let $\Omega \subset \mathbb{R}^n$ be an open set which is bounded in one direction, i.e. there exist a unit vector $e \in \mathbb{R}^n$ and constants $a<b$ such that
$$
a < x\cdot e < b \...
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Stochastic dominance $ \tanh Y\tanh Z \succeq \tanh X $
Let $X,Y,Z$ independent Gaussian r.v.'s with mean=variance. Let's denote these mean/variance parameters by $g_X,g_Y,g_Z>0$ respectively.
Set $T_1:=\tanh X$, $T_2:=\tanh Y\tanh Z$.
My question. ...
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$\sum_{j=2}^n\sum_{p=n}^\infty \frac1{p j^p} + \sum_{j=n+1}^\infty \sum_{p=2}^\infty \frac1{p j^p}<\frac1n$ for all $n\ge3.$
How do I prove the following inequality for all $n\ge3$?
$$\sum_{j=2}^n\sum_{p=n}^\infty \frac1{p j^p} + \sum_{j=n+1}^\infty \sum_{p=2}^\infty \frac1{p j^p}<\frac1n$$
I've tried to bound it with ...
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2
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Optimal constant in $L^1-L^2$ inequality on Gauss space
For a differentiable real-valued function on $\mathbb{R}^n$, denoting $\partial_i f$ for the $i$th partial derivative, we can define the functional
$$
T_n(f) = \sum_{i=1}^n \frac{1}{1 + \log(\|\...
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Behavior of the distance of the origin to an affine hyperplane intersected with the cube
Let $\theta_1 > \theta_2 > \cdots > \theta_n > 0$ be a positive sequence such that $\sum_{j=1}^n \theta_j = 1$. Let $\lambda \in (0, 1)$.
We can define
$$
F(\theta, \lambda) = \inf_{x \in [...
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How small can $|\frac{a}{b-c}| + |\frac{b}{a-c}| + |\frac{c}{a-b}|$ get for distinct positive $a,b,c$? [closed]
Let $\newcommand{\Rplus}{\mathbb{R}_+}\Rplus$ denote the set of positive reals. What is the value of
$$\inf\Big\{\Big|\frac{a}{b-c}\Big| + \Big|\frac{b}{a-c}\Big| + \Big|\frac{c}{a-b}\Big|:
a,b,c \in \...
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Is there a increasing, convex, superlinear $f$ with $c_1 f(x)y \leq f(xy)\leq c_2 f(x)f(y)$ such that $\mathbb{E}[f(X)] < \infty$?
The following version of a de la Vallée Poussin - criterion would be very helpful to me if it would be true. Can you say something about the truth value or give a reference?
Given a positive random ...
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Linear algebraic lemma in Weil II
The following linear algebraic lemma is used in Weil II to prove properties of $\tau$-real sheaves:
Lemma Let $A$ be a complex matrix and let $\overline A$ be its complex conjugate. Then the ...
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How to prove this conjectured trigonometric polynomial inequality?
Let $\mathbf{x}_1$, ..., $\mathbf{x}_4$ denote 4 distinct points in $\mathbb{R}^3$ and let $\mathbf{x} = (\mathbf{x}_1, \dots, \mathbf{x}_4)$. For $1 \leq a, b \leq 4$, with $a \neq b$, denote by $\...
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Please help me find this theorem on bounding the sum of column-wise maxima of monotone decreasing sequences
Where can I find the following theorem? I ended up with this from ChatGPT by asking questions. I am not able to Google and find anything. Any lore or bibliographic pointers would be appreciated.
...
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Concentration for Markov chain with spectral gap
Sub-Gaussian concentration for reversible Markov chains with spectral gap
Setup.
Let $(X_i)_{i\ge1}$ be a stationary, $\pi$-reversible Markov chain on a measurable space with spectral gap $\gamma>0$...
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Cardinal arithmetic inequalities according to ZF
Suppose $\kappa$, $\lambda$, $\mu$, and $\nu$ are cardinals which may or may not be ordinals. Can we prove without resorting to the axiom of choice either of the following:
$\kappa + \lambda \...
- 1,170
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Remez-type inequality
This question is about a statement on a Remez-type inequality from the paper Polynomial Inequalities on Measurable Sets and Their Applications by M. I. Ganzburg (Constr. Approx. (2001) 17: 275–306).
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Log concavity of a Gaussian function
Fix $t > 0$ and consider the map
$$
f(x) = \log \mathbb{P}\{|\sqrt{x} + Z| \leq t\},
$$
where $Z$ is a standard Normal random variable on the real line.
Is it true that $f$ is concave on the ...
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