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Complex stuff questions (2)

Markov

Member
Feb 1, 2012
149
Denote $D=\{z\in\mathbb C:|z|<1\}$

1) Given $z,w\in\mathbb C,$ prove that the angle between $z$ and $w$ equals $\dfrac\pi2$ iff $\dfrac zw$ is pure imaginary or $\overline zw+z\overline w=0$

2) Let $a,b,c\in\partial D$ so that $a+b+c=0.$ Prove that the triangle $\Delta(a,b,c)$ is equilateral.

3) Let $C>0$ be a fixed constant. Characterize all the $f$ entire functions so that for all $z\in\mathbb C$ with $|z|>1$ is $|f(z)|\le\dfrac{C|z|^3}{\log|z|}.$

4) Consider a function $f\in\mathcal H(D)\cap C(\overline D).$ If exists $a\in\mathbb C$ so that for all $t\in[0,\pi]$ is $f(e^{it})=a,$ prove that for all $z\in D$ is $f(z)=a.$

Attempts:

1) I don't know how to see the stuff of the angles, what's the way to prove it?

2) I don't see a way to work it analytically, how to start?

3) Since $f$ is entire, it has convergent Taylor series then $f(z)=\displaystyle\sum_{k=0}^\infty\frac{f^{(k)(0)}}{k!}z^k,$ now by using Cauchy's integral formula we have $\displaystyle\left| {{f}^{(k)}}(0) \right|\le \frac{k!}{2\pi }\int_{\left| z \right|=R}{\frac{\left| f(z) \right|}{{{R}^{k+1}}}\,dz},$ and $\dfrac{{\left| {{f^{(k)}}(0)} \right|}}{{k!}} \le \dfrac{{C{R^3}}}{{{R^{k + 1}}\log R}} = \dfrac{{C{R^{2 - k}}}}{{\log R}}$ this clearly goes to zero as $R\to\infty,$ but for $k\ge2,$ then $f^{(k)}(0)=0$ for $k\ge2$ so the functions are polynomials of degree 1.

4) I don't see how to do this one.
 
Jan 31, 2012
54
Denote $D=\{z\in\mathbb C:|z|<1\}$

1) Given $z,w\in\mathbb C,$ prove that the angle between $z$ and $w$ equals $\dfrac\pi2$ iff $\dfrac zw$ is pure imaginary or $\overline zw+z\overline w=0$
Hint:


Dot product of z with w must be 0.

If $z=x+iy$ and $w=u+iv$ then:

$z\cdot w=xu+yv=0$

or:

$xu=-yv$

Now, what is $\frac{z}{w}$?

(Multiply the numerator and denominator by $\bar{w}$ )
 

Markov

Member
Feb 1, 2012
149
If $z=x+iy$ and $w=u+iv$ then:

$z\cdot w=xu+yv=0$
I don't get this, why is not $z\cdot w=xu-yv$ ? Now $\dfrac{z}{w} = \dfrac{{(x + yi)(u - vi)}}{{{u^2} + {v^2}}} = \dfrac{{xu + (uy - xv)i + vy}}{{{u^2} + {v^2}}} = \dfrac{{(uy - xv)i}}{{{u^2} + {v^2}}},$ so this shows that $\dfrac zw$ is pure imaginary. Now for the converse, how do I start? I assume that $\dfrac zw$ is pure imaginary or the other one?

Can you help me with the other problems please?
 

Markov

Member
Feb 1, 2012
149
I need help with 2), and, can anybody check my work for 3) please?
 
Jan 31, 2012
54
I need help with 2), and, can anybody check my work for 3) please?

Very nice question (2) !



From the given $a,b,c\in\partial D$ we deduce: $|a|=|b|=|c|$.

Now, for any complex numbers $z$ and $w$ we wave:

$|z-w|^2+|z+x|^2=2(|z|^2+|w|^2)$ (Prove it)

Now,

$a+b=-c$

With the formula above we have:

$|a-b|^2=3|c|^2$

Similarly:

$a+c=-b$

and $|a-c|^2=3|b|^2$


$b+c=-a$

and $|b-c|^2=3|a|^2$


But, $|a|=|b|=|c|$, hence:


$|a-b|=|c-a|=|b-c|$


The end!
 
Last edited:

Markov

Member
Feb 1, 2012
149
Also sprach Zarathustra, can you help me with problem 1), I posted some questions there.

Can anybody check my work on problem 3)?
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,403