Articles of inequality

L2 Matrix Norm Upper Bound in terms of Bounds of its Column

I need to find an upper bound for a matrix norm in terms of bounds of its columns. I have a vector $\varepsilon_i(x) \in R^{n\times1} $ such that $||\varepsilon_i(x)||_2\le\gamma_0$. I also have a matrix $Z=[\varepsilon_1, \varepsilon_2, \varepsilon_3, … ,\varepsilon_N] \in R^{n\times N}$. Using the information $||\varepsilon_i(x)||_2\le\gamma_0$, can I find an upper bound for $||Z||_2$? If […]

Adding equations in Triangle Inequality Proof

Inequality to prove: $|a+b|\leq |a| + |b|$ Proof: $-|a| \leq a \leq |a|$ $-|b| \leq b \leq |b|$ Add 1 and 2 together to get: $-(|a|+|b|)\leq a+b\leq|a|+|b|$ $|a+b|\leq|a|+|b|$ What is the meaning of adding inequalities 1 and 2? How does adding these inequalities rely on the order axioms? Could someone break down this addition into […]

Three Variables-Inequality with $a+b+c=abc$

$a$,$b$,$c$ are positive numbers such that $~a+b+c=abc$ Find the maximum value of $~\dfrac{1}{\sqrt{1+a^{2}}}+\dfrac{1}{\sqrt{1+b^{2}}}+\dfrac{1}{\sqrt{1+c^{2}}}$

euler triangle inequality proof without words

today i was studying geometric inequalities and I saw this inequality $$R \ge 2r$$ unfortunately the book did not provided any prove or further explanations. So I just did a little research about it. I find that the name of inequality is euler triangle inequality and there was a simple proof about it If $O$ […]

A strong inequality from Michael Rozenberg

my question is about an inequality from Michael Rozenberg See here for the initial inequality My trying : We make the following substitution with $b\geq a $ et $c\geq a$: $A=a$ $AB=b$ $AC=c$ After simplification we get : $$\frac{1}{13+5B^2}+\frac{B^3}{13B^2+5C^2}+\frac{C^3}{13C^2+5}\geq \frac{1+B+C}{18}$$ This makes one study a much more general inequality than the initial one and apply […]

Show that $\frac{a+b}{2} \ge \sqrt{ab}$ for $0 \lt a \le b$

I have to prove that $$\frac{a+b}{2} \ge \sqrt{ab} \quad \text{for} \quad 0 \lt a \le b$$ The main issue I am having is determining when the proof is complete (mind you, this is my first time). So I did the following steps: $$\begin{align} \frac{a+b}{2} &\ge \sqrt{ab} \\ \left(\frac{a+b}{2}\right)^2 &\ge \left(\sqrt{ab}\right)^2 \\ \frac{a^2+2ab+b^2}{4} &\ge ab \\ […]

If $a^2+b^2=1$ where $a,b>0$ then find the minimum value of $a+b+{1\over{ab}}$

If $a^2+b^2=1$, where $a>0$ and $b>0$, then find the minimum value of $a+b+{1\over{ab}}$ This can be easily done by calculas but is there any way to do do this by algebra

Proving a logarithmic inequality

I’m interested why this is true: $$ \text{Considering }\forall (x,y,z) \in (1,\infty) $$ The following holds: $$\log_xy^z+\log_x{z^y}+log_y{z^x} \geq \frac{3}{2}$$ This is taken from a high school textbook of mine. I tried finding a meaningful manipulation by using AM-GM, but that got pretty messy. I’d like to avoid Lagrange multipliers since this is meant to be […]

How to prove $\left(|a+b|^p+|a-b|^p\right)^{1/p}\ge 2^{1/p}\left(a^2+(p-1)b^2\right)^{1/2}$

For real numbers $a, b$ and all $1\le p\le 2$, how to prove $$\left(|a+b|^p+|a-b|^p\right)^{1/p}\ge 2^{1/p}\left(a^2+(p-1)b^2\right)^{1/2}?$$

Find max: $M=\frac{a}{b^2+c^2+a}+\frac{b}{c^2+a^2+b}+\frac{c}{a^2+b^2+c}$

For $a,b,c>0$ and $abc=1$. Find max: $M=\frac{a}{b^2+c^2+a}+\frac{b}{c^2+a^2+b}+\frac{c}{a^2+b^2+c}$