# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

19,034 questions
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### Describe the ‘intervals’ $[a, b]$ and $(a, b)$, in the case where $a = b.$

This is a question from my textbook, however there are no solutions, would $[a,b] = a$ and $(a,b) = \text{undefined}$? I'm not sure if $(a,b)$ is right.
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### Hermite-Hadamard's integral inequality for improper integrals

Let's say that $f(x)$ convex in the interval $[b,\infty)$, write the improper integral $$\int_b^\infty f(x) dx.$$ Do Hermite-Hadamard's integral inequalities holds for this improper integral? Is ...
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### integral inequality: midpoint of interval vs. expectation over interval

I have been thinking about an inequality that should be self-evident, but which I have difficulties proving formally. It looks like the following: Take a function $f(x)$ with $f'(x)>0$ and scalars ...
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### Matrix inequality related to Minkovski space.

$M=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\end{pmatrix}$ $L\in M_4(\mathbb{R})$ s.t. $L^tML=M$ ($L^t$ is the transposition of $L$) Please ...
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### Prove $\sum\limits_{cyc}\frac{ab}{b^{\,2}+ c^{\,2}}\geqq \frac{3}{2}$

For $a\geqq b\geqq c> 0$. Prove $$\frac{ab}{b^{\,2}+ c^{\,2}}+ \frac{bc}{c^{\,2}+ a^{\,2}}+ \frac{ca}{a^{\,2}+ b^{\,2}}\geqq \frac{3}{2}$$ I used discrim to find and I want to see a solution ...
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### Prove $2\sum\limits_{cyc}\,a^{\,3}+ 3\,abc\geqq 3\sum\limits_{cyc}\,a^{\,2}b$ [on hold]

For $a,\,b,\,c\geqq 0$ and $b\equiv {\rm mid}\,\{\,a,\,b,\,c\,\}$. Prove $$2\sum\limits_{cyc}\,a^{\,3}+ 3\,abc\geqq 3\sum\limits_{cyc}\,a^{\,2}b$$ Inspried from $\lceil$ Prove $k=0$ is the only non-...
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### Norm 2 against norm inf

We know from basic linear algebra that $\forall x \neq 0, \frac{||x||_2}{||x||_{\infty}} \leq \sqrt{n}$ (where $n$ is the dimension).We also know that the equality occurs if and only if all ...
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### Given $3$ positive reals $a$, $b$ and $c$ such that $a+b+c = 1$, show that $a^a b^b c^c + a^b b^c c^a + a^c b^a c^b \le1$.

Good Day! How are you doing? I was learning about the awesome A.M - G.M. inequality from the Brilliant Wiki. There was a question in the exercises: Given $3$ positive reals $a$, $b$ and $c$ such ...
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### Is $\|\mathbf{u} -\mathbf{v}\|\leq \|\mathbf{u}\| + \|\mathbf{v}\|$?

Here's my working: $\|\mathbf{u} -\mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2 + (- 2\, \mathbf{u}\,\bullet\mathbf{v})$ Since, by the Cauchy-Schwarz theorem, \$|\mathbf{u}\,\bullet\mathbf{v}| ...