By Maurice Heins

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M y r b e r g admit n o n - c o n s t a n t S of d one K e r ~ k 3 a r t o each in q u e s t i o n The p r e i m a g e last fact t o g e t h e r on has c o n s t a n t structure , genus w i t h the o d d integers, reasons. is o b t a i n e d exists The of this c h a p t e r of infinite of the type c o n s i d e r e d This owes structure~(analytische surface for o b v i o u s S surfaces which n - cos ~z = O. center m a p has two c o m p o n e n t s , Riemann of R i e m a n n surface function. it was published.

52 - Chapter Boundary I. In this c h a p t e r classes given a Schottky our p r i n c i p a l on the border doubling, concern of a c o m p a c t we m a y and do assume the c l o s u r e of a r e g i o n ~ of a c o m p a c t of a finite number of d i s j o i n t ists a u n i v a l e n t mapping ~ onto functions (>o) anticonformal S-~. L Problems w i l l be the study of functions bordered Riemann regular consists (F) in terms Riemann surface. that our c o m p a c t surface analytic m a p ~ of S onto Our p r o g r a m in the class IV bordered Jordan itself k e e p i n g in o b t a i n i n g of functions H tions involving extension sequence S-~, Hardy classes of M .

P We turn We the e q u i v a l e n c e o f the O, M(Ifl 2) = 1, for equivalence one= < + ~. component the a i d of T h e o r e m section. is not e x c e e d i n g then the q u a s i - b o u n d e d functions. section m log(I/Ifl) at m o s t one. is for our p r e s e n t are equal, Q (b). component of this the so o b t a i n e d for h a r m o n i c the q u a s i - b o u n d e d PL f u n c t i o n s to a n a l y t i c Hence zero. sentence log F principle first p a r a g r a p h at once w i t h first and the is e s t a b l i s h e d .