Some estimates of precision of the Huygens approximation

In this paper are given some estimates of precision of the Huygens approximation $x\approx \frac{2}{3}\sin x+\frac{1}{3}\tan x,$ for right neighbourhood of zero, by determining some boundaries for the Huygens function $f(x)=\frac{2}{3}\sin x+\frac{1}{3}\tan x,$ for $x \in \left(0, \frac{\pi}{2} \right)$, in forms of some polynomial and some rational functions.