What functions have this logit-like property?

I'm interested in a family of functions $L:\mathbb{R}_+^2\rightarrow \mathbb{R}_+$ defined by a fraction of two functions $f,g:\mathbb{R}_+\rightarrow \mathbb{R}_+$

$$L(x,b)=\frac{f(x)}{f(x)+g(b)}$$

where $f'(a)>0$ and $g'(b)>0$. For some $f,g$ the function $L$ looks like a logistic curve (e.g. $f(x)=x^2$, $g(b)=b^2):$

$\hspace{1.8cm}$

The logistic curve has thi nice property of being convex in the lower portion and concave otherwise. Are there any general conditions for $L$ having this property? We may inspect the second derivative,

$$\frac{\partial^2 L(x,b)}{\partial^2 x}=\frac{g(b)\big(f''(x)[g(b)+f(x)]-2f'(x)\big)}{(g(b)+f(x))^3}.$$

Obviously, this is concave in $x$ all the way when $f''(x)<0$ (given that $f'(x)>0$). When $f''(x)>0$, $L$ has the 'logit property'

$$logit \hspace{0.1cm} property \iff \begin{cases} 2f'(x)<f''(x)[g(b)+f(x)], \hspace{0.1cm} x \in [0,x_0]\\ 2f'(x)>f''(x)[g(b)+f(x)], \hspace{0.1cm} x \in (x_0, \infty] \end{cases}$$

For what sort of functions $f,g$ is this satisfied? Is there some way of specifying some general conditions that these need to meet, beyond what has already been mentioned about the first and second derivatives?

• Are you only interested in the $x$ dependence? Why not just define $g$ as a constant, since it doesn't depend on $x$? It doesn't really matter, unless you want to avoid $f+g=0$ for some $x,b$? – Yuriy S Mar 30 '18 at 13:39