Terminology for all derivatives non-negative

Question

I have a real-valued function $g(t)=e^{f(t)}$ where $t$ is either real or complex and lies in an open interval around $t=0$. I want to prove $g(t)$ is infinitely differentiable, and furthermore that all of the derivatives at $t=0$ are non-negative. For example, this condition is satisfied for the trivial function $f(t)=t$. Is there a technical term for this non-negativity condition? And are there any common techniques for this kind of proof?

Answer 1

$f: f^{(l)}\geq 0$ is called absolutely monotone, or totally monotone; see

https://www.encyclopediaofmath.org/index.php/Absolutely_monotonic_function

Answer 2

The derivative of $g(t)$ can always be computed with the chain rule, so long as $f(t)$ is differentiable. If the function $f(t)$ has the form $f(t) = b(t+c)^a$ where $b,c$ are real constants and $1 < a < 2$, then the derivative of $g(t)$ becomes $$ g'(t) = f'(t)e^{f(t)} = ab(t+c)^{a-1}e^{f(t)}. $$ As long as $t + c > 0$, so we don't need to make sense of possibly taking the $(a-1)$th root of a negative number, for instance, then the derivative $g'(t)$ is positive. When a function's derivative is positive everywhere, that function is an increasing function in the sense that whenever $t_1 < t_2$, then $f(t_1) < f(t_2)$. Moreover, $g(t)$ is infinitely differentiable because it is the composition of two infinitely differentiable functions, one of them being $t\mapsto e^t$ and the other $t\mapsto f(t)$. You could prove this using the chain rule and induction, for instance.