Question

Complex analysis question
Let D be the open unit disc. Which of the following are correct?

Answer 1

i) There exists a holomorphic function f : D-> D with f(0)=0 and f'(0)=2
ii)There exists a holomorphic function f : D-> D with f(3/4)=3/4 and f'(2/3)=3/4
iii)There exists a holomorphic function f : D-> D with f(3/4)=-3/4 and f'(3/4)=-3/4
iv)There exists a holomorphic function f : D-> D with f(1/2)=-1/2 and f'(1/4)=1

Answer 2

@ganeshie8

Answer 3

@Loser66

Answer 4

@Kainui

Answer 5

@Jaynator495

Answer 6

@mathstudent55

Answer 7

@loser66

Answer 8

I saw this problem yesterday and I was so lazy to go over it. You can understand that I don't know how to do. hehehe

Answer 9

I have no idea of solving this type of problem. However I read a theorem "Riemann mapping theorem", let me state it here/

Answer 10

which chapter are you in?

Answer 11

If D is a simple connected region which is not the whole complex plane and let a belongs to D, then there exists a unique analytic function f: D-> C , with the property:
i) f(a)=0 and f'(0)>0
ii) f is one-one
iii) f(D)={z : |z|<1}

Answer 12

But this theorem is not applicable may be

Answer 13

so, zero of analytic function, right?

Answer 14

yeah...

Answer 15

I am sorry. I don't know. :)

Answer 16

@ganeshie8

Answer 17

@mathmate

Answer 18

@jango_IN_DTOWN
Sorry, not my field!

Answer 19

@imqwerty

Answer 20

The details escape me at the moment (I'll need a hefty review session) but I believe this is needed here:
https://en.wikipedia.org/wiki/Schwarz_lemma#Schwarz.E2.80.93Pick_theorem