What forms can a function take such its derivative is greater than or equal to the function? Following this question and discussion recently
Is the derivative of a function bigger or equal to $e^x$ will always be bigger or equal to the function ?!
I decided to look at the  different forms that a function which is always less than its derivative takes, so instead of looking at when $\frac{f'(x)}{f(x)} \ge 1$. I restated the question to this.
Let $g(x)$ be a function such that $g(x) \ge 1\space  \forall x$ then solving the following the inequality will give us the functions we need.
$$\frac{f'(x)}{f(x)} = g(x)$$
Solving that differential equation you get that $f(x)$ takes on the form (where $k$ is a constant and $g(x) \ge 1 \space \forall x$)
$$f(x) = ke^{\int g(x)dx}$$
My question is do all functions that have derivatives greater than the function itself have this form or am i missing something?
 A: We are looking for a simple criterion for a positive function $f$, which is equivalent to the differential inequality $f'(x)\geq f(x)$ for all $x$. One can say the following:
A differentiable function $f:{\mathbb R}\to{\mathbb R}_{>0}$ satisfies $f'(x)\geq f(x)$ for all $x$ iff the function $$g(x):={f(x)\over e^x}$$
is increasing.
Proof. One has $g'(x)=\bigl(f'(x)-f(x)\bigr)e^{-x}$, and this is $\geq0$ for all $x$ iff $f'(x)\geq f(x)$ for all $x$.
A: I found this condition, that probably is very similar to yours, except in the concretion of the constant $k$. Integrating the inequality from a point $x_0$ of the domain of $f$,
\begin{align}
\int_{x_0}^xdx \leq \int_{x_0}^x\frac{f'(x)}{f(x)}dx \iff x - x_0 \leq \ln\frac{f(x)}{f(x_0)} \iff f(x) \geq f(x_0)e^{x - x_0}.
\end{align}
However, about the implication $\Leftarrow$ of the first step, where I integrate both sides, I think that it's not true in general, so maybe here you have some constraint for $f$.
A: What if $g(x) = h(x,f(x))$ ? The solution is not the most general case.
I think the best thing you can get is a general differential equation.
A more general case is to take
$$ \frac{f'}{f} = 1+ g(x,f(x) , f'(x))$$
for some non-negative function $g$.
