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We as f u n c t i o n a l s over are shall linearly prove t h a t the class independent as functionals they are also l i n e a r l y of analytic boundary v a l u e s b e l o n g i n g t o t h e Holder over independent vector-functions i n D* with class. IB) i n t o | 0 " G ' V f - (2-19) 0 ( 2 . 21) o k k u ( t ) ,. . , u ( t ) also h C u ( t ) , H=m' . the are the solutions solution of this of equation equation. 22) obtain: " * V •lb J 34 " S 3 ? t a . 24) and i(t ). 25) one *(t ) - u(t ) o Hence, ( J ( t ) finds: , t er.

32) condition which and functions teD variables and on D*ur, t o be f o u n d , a n a l y t i c on D*ur, i s analytic In(1-2) with with respect satisfying is to Holder's f(z) is a given i n d o m a i n D* a n d that respect t o z branch of ( i n domain t e r and e q u a l t o z e r o a t z=0. 1 Since class. 1) t o s i n g u l a r inteqral t o zero. 54): Substituting J — r t=ct(z) Integrating both f f sides | e D ( 2 . 3 5 ) , we («(z»- 35! 35) g- with ' C e D ' J = 1 2 ' - 2 < - 3 5 ) obtain: .

9) , t h e n cj(z)=0. n with are simple analytic , . . ,ej (z) ) s a t i s f i e s Let u(z) the theorems. t h e number non-homogeneous I f of numbers. 9). assumed o v e r t h e f i e l d The are g homogeneous e q u a t i o n complex of lemma 2 . 1 . 16) r w h e r e D~ i s t h e c o m p l e m e n t o f D*u Proceeding to the limit at T t o t h e whole z-»t (zeD~) 33 complex p l a n e . 66) we obtain D(t)K(t,t ) d t + - ^ ' . 17), we find t d ( t ) =0, t e r . 1 is proved. 14)): | o(t)f(t)dt, j j = l m;.