Free boundary value problems for analytic functions in the closed unit disk

We shall prove (a slightly more general version of) the following theorem: let Phi be analytic in the closed unit disk (D) over bar with Phi :[0;1] -- > (0; 1], and let B(z) be a finite Blaschke product. Then there exists a fu nction h satisfying: i) h analytic in the closed unit disk (D) over bar, ii ) h(0) > 0, iii) h(z) not equal 0 in (D) over bar, such that F(z) := integral(o)(z) h(t)B(t)dt satisfies \F'(z)\ = Phi(\F(z)\(2)); z is an element of partial derivative D. This completes a recent result of Kuhnau for Phi(x) =1+alpha x, -1 < alpha< 0, where this boundary value problem has a geometrical interpretation, name ly that beta(alpha)F(r(alpha)z) preserves hyperbolic arc length on partial derivative D for suitable beta(alpha); r(alpha). For these important choice s of Phi we also prove that the corresponding functions h are uniquely dete rmined by B, and that zh(z) is univalent in D. Our work is related to Beurl ing's and Avhadiev's on conformal mappings solving free boundary value cond itions in the unit disk.