I made a simple model several weeks ago which led to a conjecture, which I have been unable to prove thus far.

Given constants $\displaystyle s$ and $\displaystyle h$ such that
$\displaystyle 0<s<h/2$,
functions $\displaystyle p(x)$ and $\displaystyle C_{i}(x)$ ($\displaystyle i\in\{1,2\}$) such that
$\displaystyle p(z)>0,p'(z)<0,p''(z)>0$
$\displaystyle C_{i}'>0,C_{i}''>0$
and values
$\displaystyle \exists x,y:\qquad-p'(x+y)=\frac{C_{1}'(x)}{h/2}=\frac{C_{2}'(y)}{s}$
$\displaystyle \exists x^{*},y^{*}:\qquad-p'(x^{*}+y^{*})=\frac{C_{1}'(x^{*})}{h-s}=\frac{C_{2}'(y^{*})}{s}.$

Prove or disprove that the following always holds:
$\displaystyle h(p(x+y)-p(x^{*}+y^{*}))>C_{1}(x^{*})-C_{1}(x)+C_{2}(y^{*})-C_{2}(y)$

Best attempt so far:
$\displaystyle h(p(x+y)-p(x^{*}+y^{*}))>h(x+y-x^{*}-y^{*})p'(x^{*}+y^{*})$
$\displaystyle =h(x^{*}-x+y^{*}-y)(-p'(x^{*}+y^{*}))(\frac{h-s}{h}+\frac{s}{h})$
$\displaystyle =(x^{*}-x+y^{*}-y)(C_{1}'(x^{*})+C_{2}'(y^{*}))$
I've tried using other inequalities, but I always managed to find counterexamples in Mathematica.
So far, I have not found a counter examples for $\displaystyle (x^{*}-x+y^{*}-y)(C_{1}'(x^{*})+C_{2}'(y^{*}))>C_{1}(x^{*})-C_{1}(x)+C_{2}$, so I think I'm in the right direction.
Perhaps someone knows of an application which can search for counterexamples given such conditions?

Any suggestions would be very, very, very much appreciated.