Question

THRM y a bed and f e HallD, ID) , we have disto (fa), em s distpla.b). (Schwmg-Pie demon). Exercise mma

Answer 1

وأهد i a bED and 16 HelCD. D) ditt p (fcal, f(b)] _distpconb) It is a holomorphich fernceton Define the automorphism de (a) = ana This is a Blaschke factor Note that db, maps b to o and 886) maps o to b This pfcb) o fo 06 maps o to o. Thus we can apply the schwarz - ptcle lemma to conclude that refibo fopis) call Ljal for all atD fix a Ed. Because & is an automosphism, there is a unicque a = .ca) such that Das = a mamely a = b 03 thus Idees o fraul la = 1 as desired T-bat

of fis an aitomorphism of D then the same argument as above applies to f! thus we get for all a b ED distp (f (a), f'(o)) = distp Ca, to and letting a to be feas, f(b) we get disfp Glas, Abi) < d'stpla, bj