Mixed Riemann-Hilbert boundary value problem with simply connected fibers

We study the existence of solutions mixed Riemann-Hilbert or Cherepanov boundary value problem with simply connected fibers on unit disk Δ. Let L be a closed arc ∂Δ end points ω−1,ω1 and let smooth function no zeros. {γξ}ξ∈∂Δ∖L˚ family Jordan curves in C which all contain point 0 their interiors such that γω−1, γω1 are strongly starshaped respect to 0. Then under condition for each w∈γω±1 angle between w normal γω±1 at is less than π10, there exists Hölder continuous f Δ‾, holomorphic Δ, thatRe(a(ξ)‾f(ξ))=0 Landf(ξ)∈γξ ∂Δ∖L˚.