3.3.10 Concave Up Functions

We will determine for any concave up function . Since g is concave up, . Let ; we assume that , so we may take . A simple proof by contradiction, which follows, shows that is an upper bound for g:

Assume that there is a point such that . Let , so . Furthermore, and imply that .

A quick review of the chart reveals that this situation is impossible. There is no such that since , , and .

The assumptions made do not overly restrict the applicability of the proof.

• If , consider g|[0,1] in place of g.
• If , consider in place of g, such that g' is concave up. If exists, it may be taken for y; otherwise, a trivial upper bound may be used.
• If , consider in place of g, such that g' is concave up. If exists, it may be taken for y; otherwise, a trivial upper bound may be used.

The bound is optimal, since may not be lowered. Lowering would lower or , invalidating as an upper bound.

Jeff Tupper

March 1996