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We shall prove (a slightly more general version of) the following theorem: let 'D be analytic in the closed unit disk D with ' : [0, 1] -+ (0, 1], and let B(z) be a finite Blaschke product. Then there exists a function h satisfying: i) h analytic in the closed unit disk D, ii) h(O) > 0, iii) h(z) / 0 in D, such that F(z) := h(t)B(t)dt satisfies IF'(z)I = ,(IF(z)I2), z E 0D. This completes a recent result of Kiihnau for (I(x) = 1 + ax, -1 < a < 0, where this boundary value problem has a geometrical interpretation, namely that 3(a)F(r(a)z) preserves hyperbolic arc length on 89D for suitable /3(a), r(a). For these important choices of 'I we also prove that the corresponding functions h are uniquely determined by B, and that zh(z) is univalent in D. Our work is related to Beurling's and Avhadiev's on conformal mappings solving free boundary value conditions in the unit disk.

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