Beurling's boundary differential relations on multiply connected domains

Beurling's boundary differential relations for holomorphic functions on a multiply connected domain D in C are considered. Let k >= 3. The existence result is proved for the boundary differential relations of the form vertical bar f'(xi)vertical bar = Phi(f(xi)), xi is an element of partial derivative D, where Phi is a positive C-k function on C. Moreover, the existence of holomorphic solutions is proved for rho(xi, f'(xi)) = Phi(xi, f(xi)), xi is an element of partial derivative D, where rho is a Ck+1 defining function for a family of Jordan curves in C containing the point 0 in its interior and 4, is a positive C-k bounded function on partial derivative D x C. (C) 2015 Elsevier Inc. All rights reserved.