I'm working on an economics paper, and in the model I've made I've basically gotten myself a little bit stuck. I need to show that there exists a nondecreasing concave function $u$ and numbers $P$ and $\theta$ with $P>\theta>0$, and $\gamma\in[0,1]$ such that:$u(P-\theta)-u(-\theta)>\frac{1}{1-\gamma}(u(P)-u(0))$
And in fact what I would like to show is that for any monotonically increasing $u$ which is strictly concave, we can find $P$, $\theta$, and $\gamma$ which satisfy the equality. I don't have much experience proving that kind of a statement though and I'm having some trouble getting started. Does anyone have any idea what would be a good way to start proving that statement (if it's even true—when I draw pictures of what I want it looks like it should be true but maybe it's not)?
EDIT
I've been puzzling over this and I realize that it will follow directly from a lemma: for $a>b$ and $c>0$, $u(a)-u(b)>u(a+c)-u(b+c)$, since then I can say that $u(P-\theta)-u(-\theta)>u(P)-u(0)$ just by adding $\theta$ to the arguments of $u$ on the left hand side. So I just need to prove that lemma. I can see geometrically why it must be true but I still can't quite make it follow form the concavity of $u$. I'm close though, so I may end up just answering my own question. Writing it out like this is helping.