  For a harmonic diffeomorphism between the Poincar\'{e} disks, Wan showed the
equivalence between the boundedness of the Hopf differential and the
quasi-conformality. In this paper, we will generalize this result from
quadratic differentials to $r$-differetials. We study the relationship between
bounded holomorphic $r$-differentials and the induced curvature of the
associated harmonic maps from the unit disk to the symmetric space
$SL(r,\mathbb R)/SO(r)$ arising from cyclic/subcyclic harmonic Higgs bundles.
Also, we show the equivalences between the boundedness of holomorphic
differentials and having a negative upper bound of the induced curvature on
hyperbolic affine spheres in $\mathbb{R}^3$, maximal surfaces in
$\mathbb{H}^{2,n}$ and $J$-holomorphic curves in $\mathbb{H}^{4,2}$
respectively. Benoist-Hulin and Labourie-Toulisse have previously obtained some
of these equivalences using different methods.
