Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

19,268 questions
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A Euler summation like inequality.

Let $f: \mathbb{R} \to \mathbb{C}$ be a continuously differentiable function. Let $n$ be an integer. Why do we have: $$\int_n^{n+1} f(t) dt = f(n)+O\left(\int_n^{n+1} |f’(t)| dt\right)?$$
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$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$ given that where $x, y, z > 0$ and $xyz = \frac{1}{2}$.

$x$, $y$ and $z$ are positives such that $xyz = \dfrac{1}{2}$. Prove that $$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$$ Before you complain, this problem ...
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Bound $\|(A+B)-(A^{1/4}(1+A^{-1/2}BA^{-1/2})^{1/2}A^{1/4})^2\|$ in terms of commutator $\|AB-BA\|$

For positive definite matrices $A$ and $B$, can $$\|(A+B)-(A^{1/4}(1+A^{-1/2}BA^{-1/2})^{1/2}A^{1/4})^2\|$$ be bounded in terms of $\|AB-BA\|$? Note that if the matrices commute, then both norms ...
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$x$, $y$ and $z$ are positives such that $xy + yz + zx = 1$. Prove that $$\large \frac{1}{x^2 + 1} + \frac{1}{y^2 + 1} + \frac{1}{z^2 + 1} \ge \frac{2}{3}\left(\frac{x}{\sqrt{x^2 + 1}} + \frac{y}{\... 1answer 80 views Prove inequality without using Möbius transformations Is there a simple way to prove the following inequality without using Möbius transformations: \left| \frac{{z}_{1}-{z}_{3}}{1-{z}_{3}\bar{{z}_{1}}} \right| \leq \left| \frac{{z}_{1}-{z}_{2}}{1-{z}_{... 1answer 21 views relations betweem Schaten norms Let for a matrix A \sigma(A)=(\sigma_1(A) \ldots \sigma_n(A)) be a sequence of it singular values. The p-th Schatten norm is defined as$$ \|A\|_{S_p}=\|\sigma(A)\|_p, \quad 1\leq p \leq \infty. $$... 3answers 127 views Let A, B and C be the angles of an acute triangle. Show that: \sin A+\sin B +\sin C > 2. [duplicate] Let A, B and C be the angles of an acute triangle. Show that: \sin A+\sin B +\sin C > 2. I started from considering$$\begin{align}\sin A+\sin B+\sin (180^o-A-B) &= \sin A+\sin B+\sin(A+B)...
A few days ago I asked this question, that basically, it said: If it is stated that, $-2.2\leq p \leq 2.3$. Can you say that, $|p| \leq 2.4$? This is true, because, $[2.2,2.3] \subset [-2.4, 2.4]$ ...