Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
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(Dis)Prove $\sum_{i=1}^n\sum_{j=1}^n{(|x_{i}-x_{j}|-|y_{i}-y_{j}|)^2}\geq 4$

Let $n\ge 4$ and two vectors $x$ and $y$ in $\mathbb{R}^n$ that satisfy $\sum_{i=1}^{n}{x_{i}^2}=\sum_{i=1}^{n}{y_i}^2=1$ $\sum_{i=1}^{n}{x_{i} y_i}=0$ $\sum_{i=1}^{n}{x_{i}}=\sum_{i=1}^{n}{y_i}=0$ ...
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Are these upper and lower bounds for $\frac{x!}{\left\lfloor{x}\right\rfloor!}$ useful? If so, are they already known?

Truncating the infinite series for the derivative of the Digamma function $$\psi'(x) = \sum_{n=0}^\infty\frac{1}{(x + n)^2}$$ after $m-1$ terms, where $m$ is a positive integer (the case $m=2$ ...
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Inequality involving power to fractional part of integer multiples of logarithm of integer to coprime base.

For $x \in \mathbb{R}^+$, let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. Let $k \in \mathbb{N}$. Show that $2^{\{k \log_2(3)\}} < \dfrac{2}{1 + 2^{-k}}$ for $k > 1$. ...
Gronwall's inequality says that solutions to the initial value problem $u'(t) \leq \beta(t)u(t)$ with $u(0)=u_0$ are bounded by solutions to the problem with inequality replaced with equality for $t\... 0answers 224 views Von Neumann's Trace Inequality for Multiple Matrices Von Neumann's trace inequality states that$|tr(AB)| \le \sum_{i=1}^{n} \sigma_i(A) \sigma_i(B)$where$A, B$are general square matrices with singular values$(\sigma_i(A)), (\sigma_i(B))$, ... 0answers 94 views An inequality with a constraint Let$x_1,...,x_n , y_1,...,y_m$be real numbers such that$\sum_{i=1}^n x_i=\sum_{j=1}^my_j=0$. Then how to show that for any real numbers$a_1,...,a_n $and$b_1,...,b_m$,$2\sum_{i=1}^n \sum_{j=1}...
Hello everybody I have a question about this : Let a function $f$ with domain $]0,+\infty[$ and codomain $]0,+\infty[$ and twice differentiable with the following inequality : f'+f''\geq f^2>...