# Existence of a function between two functions $f$ and $g$ where $f \in o(g)$

Is this statement true?

For each two functions $$f$$ and $$g$$, where $$f \in o(g)$$, there exists a function $$h$$ where $$f \in o(h)$$ and $$h \in o(g)$$

Please note that I am using small $$o$$ notation.

For example, let $$h=\sqrt{fg}$$, assuming $$f$$ and $$g$$ are positive.
If not, we can let $$h(x)=\max(\sqrt{|f(x)g(x)|}, g(x)/(|x|+1))$$. The term, $$g(x)/(|x|+1)$$ makes sure that $$h(x)$$ will stay positive eventually.
Exercise. For each two functions $$f$$ and $$g$$, where $$f \in \omega(g)$$, there exists a function $$h$$ where $$f \in \omega(h)$$ and $$h \in \omega(g)$$.