I'd be interested in seeing a repost of the proof. This will be, uh, a tad longer than my old posting, so here is an overview. Note that -1 1. If there is interest I could give more detail. If there is interest I could give more detail.
V is closed under the differentiation operator.
V includes sin and cos . Let B be the subset of V of functions for which all the upper bounds are at most 1. Then we are doing maximization of a continuous linear function on a convex set. I don't know very much about extremal points in convex families of functions, except to say that this line of reasoning shows up when trying to prove the existence of really useful uniformization functions, so the topic must be of interest to analysts. The Bieberbach conjecture was also of this sort, of course, and difficult to prove.