I am currently reading an article by Berk and Green (2004) "Mutual Fund Flows and Performance in Rational Markets". On page 1273, the following is written:
"We assume that these costs are independent of ability and are increasing and convex in the amount of funds under active management. Denote the costs incurred when actively managing a fund of size $q_t$ as $C(q_t)$, and assume, for all $q\geq{}0$, that $C(q)\geq{}0$, $C'(q)>0$, and $C''(q)>0$, with $C(0)=0$ and $\lim_{q\to\infty}C'(q)=\infty$.
Maybe I am missing something, but how can $C'(q)>0$ and $C''(q)>0$, when it is stated that it is for all $q\geq{}0$, which implies that $0$ is included. Shouldn't it then be written $C'(q)\geq{}0$ and $C''(q)\geq{}0$ when $q=0$?
What is written in the article is correct.
I think your confusion comes from this: How can $C'(q)>0$ and $C''(q)$ for $q=0$ when $C(0) = 0$? (And probably, you have in mind some function like this: $C(q) = q^2$.) Yet observe the following function:
$$C(q) = q^2 + q,$$ which satisfies all properties of the function in the article.