Looking for all strictly concave functions unbounded in $\mathbb{R}^+$

Question

Looking for all functions $\left.g:[k,\infty )\to \left[k',\infty \right.\right)$, with $k, k'>0$ that satisfy the following conditions:

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

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

3. $$g^{-1} (x) \ \text{is unique (g is invertible) in domain.}$$

So far I got:

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

2. $$g= \log(x)$$

Answer 1

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.

Answer 2

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.