  In a recent paper, Madritsch and Tichy established Diophantine inequalities
for the fractional parts of polynomial-like functions. In particular, for
$f(x)=x^k+x^c$ where $k$ is a positive integer and $c>1$ is a non-integer, and
any fixed $\xi\in [0,1]$ they obtained \[\min_{2\leq p\leq X} \Vert \xi \lfloor
f(p)\rfloor \Vert\ll_{k,c,\epsilon} X^{-\rho_1(c,k)+\epsilon}\] for
$\rho_1(c,k)>0$ explicitly given. In the present note, we improve upon their
results in the case $c>k$ and $c>4$.
