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 |f′(ξ)|=Φ(f(ξ)), ξ∈∂D, where Φ is a positive Ck function on C. Moreover, the existence of holomorphic solutions is proved for ρ(ξ,f′(ξ))=Φ(ξ,f(ξ)), ξ∈∂D, where ρ is a Ck+1 defining function for a family of Jordan curves in C containing the point 0 in its interior and Φ is a positive Ck bounded function on ∂D×C.