What is this economic theory? Cost of investment vs production accounting for time?
Sorry I don't have enough reputation yet in the Economics community to comment. But the answer provided by @Alecos Papadopoulos can be illustrated further. Please feel free to move my "answer" to comments of his answer. 

With the assumption of only $F' &gt; 0$ and $F &gt; 0$ and $h' &gt; 0$, then there is a simple case where the minimization problem has no solution. Consider when $F(q) = \sqrt{q}$ and $h(x) = x$. Then the objective function becomes $1 / \sqrt{q} - \sqrt{q} - c$, and if you're considering for any $q &gt; 0$, then there is no global minimizer for this (i.e. globally decreasing for all $q &gt; 0$ and hence the minimizer is $q \to \infty$. Thus, the FOC you'd derived is not true. Indeed, the condition you'd provided would render as $4 \le 1$ --- clearly impossible. 

In general, an increasing function $F$ over a linear function (i.e. say $F(x) / x$ in your case) will not necessarily be convex. And furthermore, when you subtract off the form $h \circ F$, the resulting $-h \circ F$ could again depend on the form of $F$ (which may not have any particular geometric form you could exploit). I think what would fix this model is a form such that $F(q) / q$ is convex in $q$ and $h \circ F$ is concave, so that $-h \circ F$ is convex, then the whole problem will be convex.