We consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle ${pi}{alpha}$, ${alpha}{in}(1,2]$, at infinity. We derive the exact interval for the variability of the real Taylor coefficients of these functions and we prove that the corresponding complex Taylor coefficients of such functions are contained in certain discs lying in the right half plane. In addition, we also determine generalized central functions for the aforesaid class of functions.