Sufficient conditions for the derivative of a quotient to be positive

Question

Suppose we have $g:(0,a)\to(0,b)$ and $h:(0,a)\to(0,c)$ for some positive constants $a$, $b$, and $c$. Now let $$ f(x)=\frac{g(x)}{h(x)}. $$ I am trying to determine if $$ \begin{aligned} &0<h^\prime(x)<g^\prime(x),\\ &0<h^{\prime\prime}(x)<g^{\prime\prime}(x), \end{aligned} $$ for all $x\in(0,a)$ constitute as sufficient conditions to guarantee a positive derivative for $f$ on its domain. I was thinking this could be proven true by manipulating the quotient rule for derivatives but wasn't seeing a path forward. Does anyone know of any theorems that might be helpful in trying to prove/disprove this conjecture? Thanks.

Answer 1

No, this is not a sufficient condition. Consider $a=c=1$, $b=2$ and

$$ g(x)=2h(x)=2x^2, \quad x\in(0,1). $$

Then

$$ 0<2x=h'(x)<4x=g'(x), \quad 0<2=h''(x)<4=g''(x), \quad x\in(0,1), $$

but

$$ f(x)=\frac{g(x)}{h(x)}= 2, \quad x\in(0,1),$$

so $f'\equiv 0$.