Articles of inequality

Verify that $\sqrt{2}\left\| z \right\| \ge \left|\Re(z)\right| + \left|\Im(z)\right|$

Verify that $\sqrt{2}\left\| z \right\| \ge \left|\Re(z)\right| + \left|\Im(z)\right|.$ I started off noting that $z=x+iy$ and that $Re(z)=x$ and $Im(z)=y$ Then I know that I have to square both sides, giving me $$2(x^2 +y^2) \ge x^2+2|xy|+y^2$$ I’m not sure what to do after this.

Show that $|\sin{a}-\sin{b}| \le |a-b| $ for all $a$ and $b$

I’ve recently been going over the mean value and intermediate value theorems, however I’m not sure where to start on this.

In search of easier manipulation of an inequality to prove it

Let the zigzag function defined as $$zz(x):=\left|\lfloor x+1/2\rfloor-x\right|,\quad x\in\Bbb R$$ Then for $k\in\Bbb Z$ we have that $zz(k)=0$, and $zz(\Bbb R)=[0,1/2]$, and $zz$ is increasing in any interval of the kind $[k,k+1/2]$, and decreasing in $[k+1/2,k+1]$ for $k\in\Bbb Z$. Then we define the function $$F(x):=\sum_{n=0}^\infty \frac{zz(4^nx)}{4^n},\quad x\in\Bbb R$$ Then I need to prove that $F(x)$ […]

Prove $\frac{\sqrt{(x+y)(y+z)(z+x)}}{2} \geq \sqrt{\frac{xy+yz+zx}{3}}$ for $x,y,z \geq 0$

We have geometric mean of pairwise arithmetic means on the left, which obeys the following inequality: $$\frac{x+y+z}{3} \geq \color{blue}{ \frac{\sqrt[3]{(x+y)(y+z)(z+x)}}{2} } \geq \sqrt[3]{xyz}$$ And on the right we have root-mean-square of geometric means, obeying the same inequality: $$\frac{x+y+z}{3} \geq \color{blue}{\sqrt{\frac{xy+yz+zx}{3}} } \geq \sqrt[3]{xyz}$$ This time I checked with Wolfram Alpha first, and apparently, the inequality […]

Given $r>0$, find $k>0$ such that $\sqrt{(x-2)^2+(y-1)^2}<k$ implies $|xy-2|<r $

Using the axioms, theorem, definitions of high school algebra concerning the real numbers, then prove the following: Given $r>0$, find a $k>0$ such that: $$\text{for all }x, y: \sqrt{(x-2)^2+(y-1)^2}<k\implies|xy-2|<r $$ I tried with several values given to $k$ and $r$ to find the relation between them. Suppose then $r=1$ and we choose $k$ to be […]

Cauchy – Schwarz for complex numbers

Let $z_1, . . . , z_n$ and $w_1, . . . , w_n$ be complex numbers. Show that $$|z_1w_1 + ··· + z_n w_n|^2 ≤ \sum ^n _{j=1} |z_j|^2 \sum ^n _{j=1}|w_j|^2$$ I basically tried to use the proof given for real numbers but I feel that something must be wrong. Could somebody look […]

How to solve this inequality with absolute value: $ \frac{\left|x-3\right|}{\left|x+2\right|}\le 3 $

Good morning to everyone. I have an inequality that I don’t know how to solve: $$ \frac{\left|x-3\right|}{\left|x+2\right|}\le 3 $$ I tried to solve it in this way: $$ \frac{\left|x-3\right|}{\left|x+2\right|}\le 3 \rightarrow \frac{\left|x-3\right|}{\left|x+2\right|} – 3 \le 0 \rightarrow \frac{\left|x-3\right|-3\left|x+2\right|}{\left|x+2\right|}\le 0 \rightarrow $$ Case 1: $$ \left|x+2\right| > 0 \wedge \left|x-3\right|-3\left|x+2\right| \le 0 $$ Case 1 a) […]

Find function such that $\sum y_i\le\sum x_i\Rightarrow\sum f(y_i)\le\sum f(x_i)$

What kind of a function $f$ must be to satisfy the following? If $\sum_{i=1}^{n} y_i \leq \sum_{j=1}^{n} x_j$, where $x_j, y_i \in [0,1],\forall i,j$ then $$ \sum_{i=1}^{n} f(y_i) \leq \sum_{j=1}^{n} f(x_j).$$ Any help would be appreciated. Thanks in advance! Preferably $f$ must be convex and increasing. $f$ is linear from the answer given by the […]

Inequalities – Absolute Value $|2x-1| \leq |x-3|$

$$|2x-1| \leq |x-3|$$ Answer is $$-2 \leq x \leq \frac43$$ My Question is HOW?

Solving a quadratic Inequality

My question is: Solve $$9x-14-x^2>0$$ My answer is: $2 < x < 7$ Though I know my answer is right, I want to know in what ways I can solve it and how it can be graphically represented. Thank you.