Let $z,w \in \mathbb C$ s.t $\bar z w \neq 1$ and $|z|\leq 1, |w| \leq 1$ then $ |\dfrac{z-w}{1-\bar z w}| \leq 1$  The hint is to show that $ |z-w|^2\leq |1-\bar z w|^2$ but i cant relate those 2
76516
$z\overline{w} \ne 1 $ Prove that $|\frac{w-z}{1-\overline{w}z}|\le 1$ if $|z|\le 1$ and $|w|\le 1 $. How to proceed this question?
75525
I've only been able to resolve this far  $$ d \mid n \Rightarrow \exists l \in \mathbb{Z}\; \textrm{such that} \; n=dl \Rightarrow x^{q^n} - x = x^{q^{dl}} - x.$$  I can't move forward.
74920
Suppose that $0\neq x\in\mathbb{R}$ and $x + \frac1x\in\mathbb{Z}$. Prove that, for all  $n\ge1$, $x^n + \frac1{x^n}\in\mathbb{Z}$.  I can't figure out and understand the question. Can you give me some hints ?
10198
Prove if $|z| &lt; 1$ and  $ |w| &lt; 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| &lt; 1$Given that $|1-zw^*|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)$I think the first part can be proven by saying $|1-zw^*| = 0$ if and only if $zw^*$ = 1.  And given the conditions that cannot be true. However I don't know if this part is right.
26982
I would like to show that  $$\forall n\in\mathbb{N}^*, \quad \sqrt{\frac{n}{n+1}}\notin  \mathbb{Q}$$  I'm interested in more ways of proofing this.  My method :  suppose that $\sqrt{\frac{n}{n+1}}\in  \mathbb{Q}$ then there exist $(p,q)\in\mathbb{Z}\times \mathbb{N}^*$ such that $\sqrt{\frac{n}{n+1}}=\frac{p}{q}$ thus   $$\dfrac{n}{n+1}=\dfrac{p^2}{q^2} \implies nq^2=(n+1)p^2 \implies n(q^2-p^2)=p^2$$ since $p\neq q\implies p^2\neq q^2$ then  $$n=\dfrac{p^2}{(q^2-p^2)}$$  since $n\in \mathbb{N}^*$ then $n\in  \mathbb{Q}$   I'm stuck here and I would like to see different ways to prove $\sqrt{\frac{n}{n+1}}\notin  \mathbb{Q}$
114960
let $a,b,c\in \mathbb{R}$, if such  $$\dfrac{1}{bc-a^2}+\dfrac{1}{ca-b^2}+\dfrac{1}{ab-c^2}=0$$ show that $$\dfrac{a}{(bc-a^2)^2}+\dfrac{b}{(ca-b^2)^2}+\dfrac{c}{(ab-c^2)^2}=0$$  Does this problem has nice methods? My idea:let $$(ca-b^2)(ab-c^2)+(bc-a^2)(ab-c^2)+(bc-a^2)(ca-b^2)=0$$ then I can't. Thanks
56179
Problem :  Let $z \in\Bbb{C}^{\times}$ such that $|z^3+\frac{1}{z^3}|\leq 2$. Prove that $|z+\frac{1}{z}|\leq 2$.  My approach : Since : $(a^3+b^3)=(a+b)^3-3ab(a+b)$ $\Rightarrow z^3+\frac{1}{z^3}=(z+\frac{1}{z})^3-3(z+\frac{1}{z})$ Now I don't know further about this problem whether or not this will help here. Please guide to solve this problem will be of great help, thanks.
66072
Consider $\bar{C} = C_{d}^0[0,1] $ is the space of continuous decreasing functions, s.a. $f(1) = 0$. We want to determine $\displaystyle \inf_{f \in \bar{C}} \sup_{x\in [0,1]} \frac{x f(x)}{\int_0^1 f(t) dt}$.   I thought about $\displaystyle xf(x) = xf(x) + 1\cdot f(1) = \int^x_1 f(t)dt - \int_1^x tf'(t)dt$. Actually I even don't know whether derivate exists or not.  Now we should consider $\dfrac{\int^1_x \left[tf'(t) + f(t)\right] dt}{\int_0^1 f(t)}$.  Then I thought about Poincare inequality, but fails there. Any hints ?
60943
If $a,b,c &gt; 0$ are such that $abc=1$, then   $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c. $$   I would be pleased if you give me a hint. Thanks in advance.
66462
I am trying to solve the following two problems:  1) Prove that the functions $\cos(z)$, $\sin(z)$ are surjective over the complex numbers.  2) Find all $z \in \mathbb C$: $cos(z) \in \mathbb R$ and find all $z \in \mathbb C$: $\sin(z) \in \mathbb R$.  For 1),I've tried to prove it for the function $\cos(z)$ (I suppose the other one is analogue)  so I've used the fact that $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$.  Let $w \in \mathbb C$, I want to show there exists $z \in \mathbb C : f(z)=w$, i.e., $\dfrac{e^{iz}+e^{-iz}}{2}=w$, multiplying by $2$ and then by $e^{iz}$ yields $e^{2iz}+1=2we^{iz}$ iff $e^{2iz}-2we^{iz}+1=0$.   I don't know if this approach is the correct one but here I've replaced $e^{iz}$ by $x$, so the solutions of the equation would be the roots of the polynomial $p(x)=x^2-2wx+1$, by the quadratic formula, I get that $x \in \{\dfrac{2w+w_0}{2},\dfrac{2w-w_0}{2}\}$, where $w_0^2=4w^2-4$.  Then, $e^{iz} \in \{\dfrac{2w+w_0}{2},\dfrac{2w-w_0}{2}\}$. At this point I got lost, I would like to explicitly show that $z$ exists and I can't see existence directly from the fact that $e^{iz}=\dfrac{2w+w_0}{2}$ or $e^{iz}=\dfrac{2w-w_0}{2}$.  First I thought of taking logarithm of both sides of the equation ir order to solve for $z$, but this is not a legitimate operation unless $e^{iz} \in \mathbb R$.   I couldn't go any farther.  For point 2) I have no idea what to do, should I use the identity $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$?  I would appreciate some help with the two points (specially with point 2), at least in 1) I could do something).   Btw, Happy new year!
68929
Let $x,y,z&gt;0$: Prove that:   $\frac{1}{(x+2y)^2}+\frac{1}{(y+2z)^2}+\frac{1}{(z+2x)^2} \geq\frac{1}{xy+yz+zx}$  I tried to apply Cauchy - Schwarz's inequality but I couldn't prove this inequality!
38666
$\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$, $|z|&gt;1$.  There are two facts that my professor uses that I am confused about.  The first is: $|1+z^n| \geq ||z|^n-1|$, I believe this is true for any $|z|$.  The other is: $\frac{1}{|z|^n-1} \leq \frac{2}{|z|^n}$, I believe this is also true for any $|z|$.  Can anyone prove these statements for me?
44025
Can someone of you give me a solution for this?     Let $N\in \mathbb N$.   a) We define the map $\mathfrak F:(\mathbb C^N, ||\cdot||_2)\to(\mathbb C^N, ||\cdot||_2)$ by $$(\mathfrak F(x))_k := \frac{1}{\sqrt N}\sum_{j=1}^Nx_je^{2\pi i\frac{(j-1)(k-1)}{N}} \quad \forall k\in \{1,...,N\}$$    Show that $\mathfrak F$ is norm-preserving, i.e. $$||(\mathfrak F(x))_k||_2 = ||x||_2 \quad \forall k\in \{1,...,N\}$$    b) For $n,m \in \{1,2,...,N\}$ we define the entries of $M\in \mathbb C^{N\times N}$ by $$M_{nm} :=  \begin{cases} \frac{N+1}{2}, \quad \quad n=m \\ \frac{1}{e^{2\pi i \frac{m-n}{N}}-1}, \,\,n \neq m \end{cases}$$   Show that $1, 2, 3, ..., N$ are the Eigenvalues of M.
35983
1) How do I prove the following:   Let A = {6a + 4b ∈ Z : a, b ∈ Z} and B = {2a ∈ Z : a ∈ Z}. Show that A = B.  Thank you all for the help!
122888
This is a problem from my complex analysis textbook.  The hint is to consider $g(z)=(z-iR)^2(z+iR)^2 f(z)$ and to show that $|g(z)| \leq 8R^2$.  This is what i have tried: Consider $Re z \geq 0$, then $|(z-iR)^2 f(z)|\leq \left(\frac{|z-iR|}{Re z}\right)^2\leq 2$   But then I don't know what to do next. Any help I'd appreciate :)
65722
I´m stuck with the section $c)$ of the following Hartshorne´s exercise  I have just proved $a)$ and $b)$. I wonder if you can help me. Thank you for your time
60079
Question 1: Let $z$ and $w$ be complex numbers satisfying $|z| = 4$ and $|w| = 2$. Then enter in the numbers $$ |z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2 $$ below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter in a question mark.  I plugged in $4$ and $2$ to each equation and got $36$, $64$, $4$, $4$, for question one. It was wrong.  Question 2: Let $z$ and $w$ be complex numbers satisfying $|z| = 5, |w| = 2,$ and $z\overline{w} = 6+8i.$ Then enter in the numbers $$ |z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2 $$ below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter in a question mark.  I don't know how to even start on question 2 because of the $z\overline{w}$.