TITLE: Is it possible to find two differentiable functions such that the following inequalities hold?
QUESTION [0 upvotes]: Let $\gamma > 0.$ Is it possible to find two differentiable functions $f, g:[0,T] \to \mathbb{R}$ such that
$$g(T)e^{-\gamma f(T)} \geq g(0)e^{-\gamma f(0)}$$
and
$$\gamma f'(t) - \frac{g'(t)}{g(t)} \geq C > 0\quad\text{for almost every $t \in [0,T]$}$$
for some constant $C >0$?

REPLY [1 votes]: Yes. $g(x) := -1$ and $f(x) := \frac{x}{\gamma}$ will do (with $C = 1$).
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