Boundary value problems for holomorphic functions on the upper half-plane

Let Pi subset of C be the open upper half-plane and let {gamma(z)}(z is an element of partial derivative Pi) be a smooth family of smooth Jordan curves in the complex plane C parametrized by the boundary of H. Then there exists a smooth up to the boundary holomorphic function f on Pi such that f(z) is an element of gamma(z) for every z is an element of partial derivative Pi. Similar result is also proved on an arbitrary bordered Riemann surface.