In this talk we will discuss two results on the multiplicity problem for prime closed orbits of dynamically convex Reeb flows on the boundary of a star-shaped domain in R2n. The first result asserts that such a flow has at least n prime closed Reeb orbits, improving the previously known lower bound by a factor of two. The second main theorem is that when, in addition, the domain is centrally symmetric and the Reeb flow is non-degenerate, the flow has either exactly n or infinitely many prime closed orbits. This is a joint work with Basak Gurel and Viktor Ginzburg.