Remark 3

If $f$ and $g$ are comparable of order $k$ this need not be so for $hf$ and $hg$ even for a monotone function $h$ as simple as $h(x) = x$ (V, p. 260, exerc. 3); likewise, $1/f$ and $1/g$ are not necessarily comparable of order $k$ (V, p. 259, exerc. 1).