Looking for all strictly concave functions unbounded in $\mathbb{R}^+$ Looking for all functions $\left.g:[k,\infty )\to \left[k',\infty \right.\right)$, with $k, k'>0$ that satisfy the following conditions:


*

*$$g'(x)>0, g''(x)<0 \ \text{for all $x$ in domain}$$ 

*$$\lim _{x\to \infty }g(x)=+\infty.$$

*$$g^{-1} (x) \ \text{is unique (g is invertible) in domain.}$$
So far I got: 


*

*$$g=x^\alpha, 0<\alpha<1.$$

*$$g= \log(x)$$
 A: If $\phi$ is any positive function, say on $(1,\infty)$ which decreases strictly to a positive number and $g(x)=\int_0^{x} \phi (y) \, dy$ then $g$ has the desired properties. There are plenty of functions $\phi$ with these properties. 
A: It is not hard to see, that for a function $g$ satisfying your assumptions $g'' \in L^1([k, \infty))$ is a necessary criterion. 
Also since $g'>0$ is requested, we can conclude by the fundamental theorem of calculus, that all $g$ satisfying property 1 have derivative of the form 
$c_1 + \int_{k}^x \phi(t)dt$, where $\phi \in L^1([k, \infty))$, $\phi <0$ and $c_1$ such that $c_1> -\int_k^x\phi(t) dt$ for all $x \in [k,\infty)$.
We have that for every $\phi \in L^1([k, \infty))$, $\phi <0$ there exists a semibounded interval $I_\phi$ of $\mathbb R$ such that $c_1 + \int_k^x\phi(t) dt > 0$ for all $x \in [k,\infty)$ and all $c_1 \in I_\phi$. You have that all functions satisfying property 1 have a derivative of the form $x\mapsto c_1 + \int_{k}^x \phi(t)dt$ for $\phi \in L^1([k, \infty))$, $\phi <0$, $c \in I_\phi$.
For property 2 to hold you simply need to choose a subset of $I_\phi$, say $\tilde{I}_\phi$ such that 
$$
x\mapsto c + \int_{k}^x \phi(t)dt \not \in L^1([k,\infty))
$$
for all $c \in \tilde{I}_\phi$
Now all functions you ask for are of the form 
$$
x\mapsto c_1 + c_2 x +  \int_{k}^x\int_{k}^y \phi(t)dt dy
$$
for any $c_1 \in \mathbb R$, any $\phi \in L^1([k, \infty))$, $\phi <0$, and any $c_2 \in \tilde{I}_\phi$.
I am aware that my solution essentially moves the whole problem into finding $\tilde{I}_\phi$ but it shows that all requested functions can be parametrized by an integrable negative function and two real numbers.
