# Questions tagged [inequalities]

for questions involving inequalities, upper and lower bounds.

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### How to estimate an integral by the variation and upper buond of the integrand?

Suppose that $f$ is a continuous function on $\mathbb{R}$. I want to estimate the definite integral $$I:= \int_{0}^a [f(x)-f(0)]dx$$ by the upper bound $M = \sup_{x\in[0,a]}|f(x)|$ and the variation ...
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### A one-sided/monotone version of min/max-stable distributions -- does this have a name?

In a couple of papers I am working on (in random graph theory) I have encountered the following property of certain probability distributions, which I will describe shortly, and I am wondering if this ...
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### Ask for a proof of an inequality involving the Bernoulli numbers

Let $B_k$ be the Bernoulli numbers and let \begin{equation} T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1. \end{equation} Prove the inequality \begin{equation*} \frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...
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### On the weighted Hilbert inequality a la Montgomery and Vaughan

The weighted Hilbert inequality estimates the norm of the skew-hermitian matrix $$B :=\left[ \frac{\delta_m^{1/2} \delta_n^{1/2}}{x_m-x_n} \right]_{m,n}$$ on $\ell_2$, where $x_1,\ldots$ is a finite ...
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### Lemma about the weighted interpolation inequality

In this article Interpolation inequalities with weights Chang Shou Lin the following lemma is stated and proved. Lemma: Suppose $\dfrac{1}{p}+\dfrac{\alpha}{n} > 0$, then there exists a constant $C$...
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### Short sequence beats long sequence

I have encountered some comparison between two binomial sums. It was amusing how the one with "fewer" summands exceeds (in value) than the other which consists of many more terms. In fact, ...
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### Techniques for solving linear inequalities

For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...
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### The lower bound of bivariate normal distribution

Suppose $(Z_1, Z_2)$ is the zero-mean bivariate normal distribution with covariance $\left( \begin{matrix} 1 & \rho; \\ \rho & 1\end{matrix} \right)$ with positive $\rho > 0$. What I want ...
Let $X$ be a smooth, complex, projective, minimal surface of general type, i.e. the canonical (line) bundle $K_X$ is big and nef. It is known that $3c_2\geq c_1^2$ (the Bogomolov-Miyaoka-Yau ...
Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$). ...