For every k in (0,1), there exists a bounded globally univalent harmonic map f = h + conjugate(g) from the unit disk to C satisfying |g'| ≤ k|h'| with h unbounded, built from a logarithmic spiral on a strip plus perturbation.
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A bounded globally univalent quasiconformal harmonic map whose analytic part is unbounded
For every k in (0,1), there exists a bounded globally univalent harmonic map f = h + conjugate(g) from the unit disk to C satisfying |g'| ≤ k|h'| with h unbounded, built from a logarithmic spiral on a strip plus perturbation.