Articles of inequality

show operator norm submultiplicative

We had in our lecture on numerical analysis the following: Let $\mathrm{Lin}(X,Y)$ be the set of all linear maps $X\rightarrow Y$. Let $A\in\mathrm{Lin}(\mathbb R^l,\mathbb R^n)$ and $B\in\mathrm{Lin}(\mathbb R^n,\mathbb R^m)$ and $\|C\|_{op}:=\max_{\|x\|\leq1}\|C(x)\|$. Then our lecturer followed $\|A\circ B\|_{op}\leq\|A\|_{op}\cdot\|B\|_{op}$. So he didn’t prove it and so I’ve tried it by my own. My attempt: $$ \|A\circ B\|_{\mathrm{op}}=\max_{\|x\|<1}\|(A\circ […]

Normed Linear Space – maximum norm vs. $||f||_1$

For $f$ in $C[a,b]$ define $$|| f ||_1 =\int_a^b |f|.$$ a. Show that this is a norm on $C[a,b]$. b. Show that there is no number $c \geq0$ for which $$||f||_{max} \leq c ||f||_1 \ for \ all \ f \ in \ C[a,b]$$ c. Show there is a $c\geq 0$ for which $$||f||_1 \leq […]

Prove that $\frac{n+1}{2} \leq 2\cdot\sqrt{2}\cdot\sqrt{2}\cdot\sqrt{2}\cdots\sqrt{2}$

Prove that : $\dfrac{n+1}{2} \leq 2\cdot\sqrt{2}\cdot\sqrt[3]{2}\cdot\sqrt[4]{2}\cdots\sqrt[n]{2}$. I am unable to prove this even by induction and general method. Indeed, when I look at the question $2\cdot\sqrt{2}\cdot\sqrt[3]{2}\cdot\sqrt[4]{2}\cdots\sqrt[n]{2}\leq n+1$, asked by me, I have received a hint as a comment to use binomial theorem and showed $$\left(1+\frac1n\right)^n=\sum_{k=0}^n{n\choose k}\frac1{n^k}\geq1+{n\choose 1}\frac1n=2.$$ So, expression becomes $$2 \cdot \sqrt{2} \cdot \sqrt[3]{2} […]

If $a+b+c=0$ and $\{a,b,c\}\subset$ so $\sum\limits_{cyc}\sqrt{1+a+\frac{7}{9}b^2}\geq3$

Let $a+b+c=0$ and $\{a,b,c\}\subset[-1,1]$. Prove that: $$\sqrt{1+a+\frac{7}{9}b^2}+\sqrt{1+b+\frac{7}{9}c^2}+\sqrt{1+c+\frac{7}{9}a^2}\geq3$$ I tried Holder and more, but without success.

Inequality between volume and its projections

Let $A \subset \mathbb{R}^3$ be connected and let’s define $A_1, A_2, A_3 \subset \mathbb{R}^2$ as projections of $A$ onto three perpendicular (to each other) planes. Show that: $$|A| \le \sqrt{|A_1| |A_2| |A_3|}\;,$$ where $|\cdot|$ is volume when applied to $A$ and area whenas $A_{1,2,3}$.

Show that ${n \choose k}\leq n^k$

Let $n$ et $k\in \mathbb{N}$ such that : $k\leq n $ Show that :$${n \choose k}\leq n^{k}$$ My thoughts: note that for all $\ k\leq n$ : $${n \choose k}=\frac{n!}{k!(n-k)!}$$ To prove that the following statement, which we will call $P(n)$, holds for all natural numbers n:$${n \choose k}\leq n^{k}$$ so my proof that P(n) […]

Beautiful cyclic inequality

Prove that cyclic sum of $\displaystyle \sum_{\text{cyclic}} \dfrac{a^3}{a^2+ab+b^2} \geq \dfrac{a+b+c}{3}$ , if $a, b, c > 0$ I’m really stuck on this one. Tried some stuff involving QM> AM(because the are positive) but can’t derive the needed ,can’t proceed from it.

Does the Schur complement preserve the partial order

Let $\begin{bmatrix} A_{1} &B_1 \\ B_1' &C_1 \end{bmatrix}$, $\begin{bmatrix} A_2 &B_2 \\ B_2' &C_2 \end{bmatrix}$ be symmetric positive definite matrices and be conformably partitioned. If $\begin{bmatrix} A_{1} &B_1 \\ B_1' &C_1 \end{bmatrix}-\begin{bmatrix} A_2 &B_2 \\ B_2' &C_2 \end{bmatrix}$ is positive semidefinite, is it true $(A_1-B_1C^{-1}_1B_1')-(A_2-B_2C^{-1}_2B_2')$ also positive semidefinite? Here $X'$ means the transpose of $X$.

How to prove the inequality $ \frac{1-e^{-x^2}}{x^2}e^{-(x-n)^2}<\frac{2}{n^2}$ for $0<x<n$

Can somebody prove that this inequality is true for $0<x<n$? $$ \frac{1-e^{-x^2}}{x^2}e^{-(x-n)^2}<\frac{2}{n^2}$$ I’m pretty much stuck.

Extending the ordered sequence of 'three-number means' beyond AM, GM and HM

I want to create an ordered sequence of various ‘three-number means‘ with as many different elements in it as possible. So far I’ve got $12$ ($8$ unusual ones are highlighted): $$\sqrt{\frac{x^2+y^2+z^2}{3}} \geq \color{blue}{ \frac{\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}}{3 \sqrt{2}}} \geq $$ $$\geq \color{blue}{\frac{\sqrt{(x+y)^2+(y+z)^2+(z+x)^2}}{2 \sqrt{3}} } \geq \frac{x+y+z}{3} \geq $$ $$ \geq \color{blue}{ \frac{\sqrt[3]{(x+y)(y+z)(z+x)}}{2} } \geq \color{blue}{\sqrt{\frac{xy+yz+zx}{3}}} \geq $$ $$\geq […]