A bounded globally univalent quasiconformal harmonic map whose analytic part is unbounded

We construct, for every $0<k<1$, a bounded globally univalent harmonic mapping $$f=h+\bar{g}\text{\hspace{0.17em}⁣}:\text{D}\to \text{C}$$ such that $$\mathrm{\mid }{g}^{\mathrm{\prime }}(z)\mathrm{\mid }\le k\mathrm{\mid }{h}^{\mathrm{\prime }}(z)\mathrm{\mid },z\in \text{D},$$ while the analytic part $h$ is unbounded. The construction is based on a bounded logarithmic spiral map on a far right horizontal strip, together with a smaller logarithmic perturbation.