  For a univalent smooth mapping $f$ of the unit disk $\ID$ of complex plane
onto the manifold $f(\ID)$, let $d_f(z_0)$ be the radius of the largest
univalent disk on the manifold $f(\ID)$ centered at $f(z_0)$ ($|z_0|<1$). The
main 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-preserving
harmonic function $f=h+\overline{g}$. The main result includes sharp upper and
lower bounds for the quotient $d_f(z_0)/d_h(z_0)$, especially, for a family of
locally univalent $Q$-quasiconformal harmonic mappings $f=h+\overline{g}$ on
$|z|<1$. In addition, estimate on the radius of the disk of convexity of
functions belonging to certain linear invariant families of locally univalent
$Q$-quasiconformal harmonic mappings of order $\alpha$ is obtained.
