Questions tagged [inequalities]
for questions involving inequalities, upper and lower bounds.
1,872 questions
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How small can $|\frac{a}{b-c}| + |\frac{b}{a-c}| + |\frac{c}{a-b}|$ get for distinct positive $a,b,c$?
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 \...
- 1,172
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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\},
$$
where $Z$ is a standard Normal random variable on the real line.
Is it true that $f$ is concave on the ...
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Inversion of conjugate to cumulant generating function?
$\newcommand{\eps}{\varepsilon}$Let $X$ be a mean-zero scalar-valued random variable with cumulant generating function $f(t) = \log \mathbb{E} e^{t X}$, where $t \in \mathbb{R}$. Let $f^\ast$ denote ...
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A general result on the convex function
During my research, I came a cross this question :
Let $f \in C([0,1])$ convex.
Is it true that $$\int_0^1 \max(f(t),f(1-t)) \geq\\ 2/3\max(f(1/3),f(2/3))+1/3 \max(f(1/6),f(5/6))?$$
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Asymptotics of a Gaussian integral
Let us consider a sequence of iid, standard Gaussian random variables $\{X_i\}_{i\geq 1}$.
Let $Y_n = \max_{2 \leq i \leq n} |X_i|$. I am interested in the asymptotic behavior of
$$
E_n(t) = \mathbb{E}...
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Cumulants and { concentration / deviation } inequalities
In some recent reading, I was reminded of the following (trimmed) quote from Terry Speed (from Cumulants and partition lattices, Australian Journal of Statistics 25(2) (1983),
378–388.)
In a sense ...
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Discrete inequality about sum of fractional parts function
I found a hard problem on an old olympiads handouts (without solution).
For an integer $n\geqslant 2$, and two positive integers $r_1$ and $r_2$ coprime to $n$, the following inequality holds: $$ \...
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An inequality involving reciprocals of angle bisectors in a triangle [closed]
Let $ABC$ be a triangle with side lengths $a,b,c$ and internal angle bisectors $\ell_a,\ell_b,\ell_c$ corresponding to vertices $A,B,C$, respectively.
I am interested in the following inequality, ...
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