Explicit expression for the extremal complex-valued kernel α-harmonic mapping U(z)

Determine an explicit closed-form expression for U(z), the complex-valued kernel α-harmonic mapping of the unit disk (for α > −1) that attains equality in the Heinz-type inequality for such mappings. The function U(z) is defined via the Poisson-type integral with boundary data given by a piecewise constant angle function θ(φ) taking values 0 on [0, π/3), 2π/3 on (π/3, π), 4π/3 on (π, 5π/3), and 0 on (5π/3, 2π).

Background

The paper establishes a Heinz-type lemma for complex-valued kernel α-harmonic self-mappings of the unit disk and proves that the lower bound 27/(4π^2) is sharp. To show sharpness, the authors construct an extremal function U(z) via a Poisson-type integral representation using a specific piecewise constant boundary function.

Although U(z) is defined implicitly by the Poisson integral representation and is shown to achieve the sharp inequality, the authors explicitly state that they do not know how to write down a direct closed-form expression for U(z), leaving this as an unresolved question.