# Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

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### Monotonicity of averages for positive-definite kernels

Let $\kappa:\mathbb{R}^d\times \mathbb{R}^d \to\mathbb{R}$ be a positive semidefinite kernel. If $A,B \subseteq \mathbb{R}^d$ are disjoint sets of equal, finite measure, then applying the definition ...
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### upper and lower bounds on rowlands sequence

rowlands sequence is defined as follows $$a_{n}=a_{n-1} + b_{n}$$ where $b_{n} = gcd(a_{n-1}, n)$ for $n>h$ it originates from E. Rowlands 2008 paper "A Natural ...
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### More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$

Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let $$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
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### Inequality involving random vectors and absolute values

Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
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### Dimensionality reduction for total variation

Let $P_i,Q_i$, $i\in[n]$, be distributions on a finite set $\Omega$. We will use $P^\otimes_{i\in[n]}$ to denote $n$-fold products of distributions. For each $i\in[n]$, define the dimensionally-...
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### A non-standard inequality for univalent functions

Related to my other question, here is an inequality from Rakhmanov's paper upon which the proof hinges. Let $F(z) = z + a_0 + \mathcal O(z^{-1})$ be analytic and univalent on $|z|>1$, continuous up ...
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### Distance between a Hölder function and a Sobolev ball

Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively. My question ...
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### Optimizing a matrix quadratic form with respect to Loewner order

Fix integers $1 \leq k \leq n$. Let $P \in \mathbb{R}^{n \times k}$ be such that $P^T P$ has full rank. Let $\mathcal{X}$ denote the set of unit trace, real $n \times n$ symmetric positive ...
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### Question on a min inequality

Is it true that $$\min\left(a^2 + b^2 - \sqrt{a^4 + b^4 + 2a^2b^2\cos(x)}, b^2 + c^2 - \sqrt{b^4 + c^4 + 2b^2c^2\cos(x-y)}, a^2 + c^2 - \sqrt{a^4 + c^4 + 2a^2c^2\cos(y)}\right) \leq \frac{1}{3}$$ ...
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### Moments on the Stiefel manifold

Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$. Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...
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### The reciprocal of the normalized tail of the Maclaurin power series expansion of the hyperbolic sinc function is a convex function

The classical Bernoulli numbers $B_j$ are generated by \label{Bernoulli-No-Generating} \frac{x}{\operatorname{e}^x-1}=\sum_{j=0}^\infty B_j\frac{x^j}{j!}=1-\frac{x}2+\sum_{j=1}^\infty ...
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### Multiplicative approximation for a negative moment of the binomial distribution

Let $X$ be a binomial random variable with parameters $n,p$. Define the function $f(n, p, t) = E\frac{1}{1 + t X},$ where $t > 0$. Question: Can we find an elementary function $F(n, p, t)$ such ...
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### Lower bound for the rank of the sum of $n$ matrices

I found a mathematical note by George Marsaglia entitiled "Bounds for the rank of the sum of two matrices", where he proves the following result. Let $A_1$ and $A_2$ be two complex matrices ...
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