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

Problem:

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.