Exercise 2, § 1

Let $\varphi$ be a strictly positive function, defined and increasing for $x > 0$, and such that $\varphi \gg 1$.

a) Show that if the function $\log \varphi(x)/\log x$ is increasing, then the relation $f \ll g$ between functions $> 0$ implies that $\varphi \circ f \ll \varphi \circ g$ if $g \gg 1$.

b) Give an example where $\log \varphi(x)/\log x$ is decreasing, $f$ and $g$ are two functions $> 0$ such that $g \gg 1$ and $f \sim g$, but $\varphi \circ f$ is not equivalent to $\varphi \circ g$.