Bounds of Analytic Functions of Two Complex Variables in Domains with the Bergman-silov Boundary.

for (Z',Xk') $ (Zk , Xk"), A , Zk < 1 (lb). For every point (Z1,Z20), ZK0 = hKk(ZkoXko) ZkAol < 1, and for every sufficiently small neighborhood U of this point, there exists o> 0 such that the set of points of b3 which belong to U consists only of points (1) for which Zk-Zkol < a and XXkol < a hold, (lc). For a fixed k and Xk, the corresponding set of points (1) is called a lamina of ek3 and is designated by ak2(k). The set a2 of points (1) corresponding to the values Al = 1, k = 1, ..., n, constitutes the so-called Bergman-Silov boundary surface of 0 on which the maximum principle is valid for functions f(z1,z2) holomorphic in 0 and continuous in $3 (see ref. 1). Let (,5o2 designate an analytic surface of the form ZK = g9(j), C 15, where Z is a domain in the c-plane and gK are functions regular in Z and continuous in Z. We assume that 5o2 has common points with Q3 and its whole boundary lies in e\Z1. Concerning the intersection of (5o2 with £, we list the following properties to be assumed as indicated in the various Theorems 1-4). The intersection 052 = 3o2 n V8 has the following representation 52 = {ZKl ZK2 = 9K() < 1}, (2a). The boundary curve gl of 52 is simultaneously the intersection 0o2 with b3, (2b). gl = ZK ZK = 9K(et"), (p C (0,27r)} can be divided into J parts: gj1 = {ZKI ZK Me = 9Kp) (C (j) (P+l)}, j= 1, ... ., J. sl < S02 < *.. < Kj+l = p1+ 2r, so that gjl c ekj3, 1 < k; < n, ki1l k,2 if il $ i2, and only the points (g1(et'i), g2(e"'i)) belong to a2, (2c). Every point of gjl lies in a certain lamina, say akj2(Xkj). Hence, by (2) functions Xkj = Xkj*()O), Zkj = Zkj*(,O), (p C (pjl fpj+1), exist such that gjl = {ZK ZK = hKkj(Zkj*(P), Xkj*(P)), ( C ('Pj, ,j+l)}I We assume that Xkj*(P) and Zkj*QP) are continuous in ((pj, oj+i), (2d). Xkj*(p) are monotone in ((j, j+l) and Xkj'(p) . 1/Q, Q > 0, p E£ (sojyj+l)X j = 1, ..., J. (2e). The expressions 1 Zkj*() go to zero not faster than some positive power of (p so;or o (pj+l if so-0; + or