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For a univalent smooth mapping $f$ of the unit disk $\ID$ of complex planeonto the manifold $f(\ID)$, let $d_f(z_0)$ be the radius of the largestunivalent disk on the manifold $f(\ID)$ centered at $f(z_0)$ ($|z_0|<1$). Themain aim of the present article is to investigate how the radius $d_h(z_0)$varies when the analytic function $h$ is replaced by a sense-preservingharmonic function $f=h+\overline{g}$. The main result includes sharp upper andlower bounds for the quotient $d_f(z_0)/d_h(z_0)$, especially, for a family oflocally univalent $Q$-quasiconformal harmonic mappings $f=h+\overline{g}$ on$|z|<1$. In addition, estimate on the radius of the disk of convexity offunctions belonging to certain linear invariant families of locally univalent$Q$-quasiconformal harmonic mappings of order $\alpha$ is obtained.

Radii of covering disks for locally univalent harmonic mappings