
Where did I go wrong
Suppose I have an odd, increasing function h with h(0)=0 and an unknown increasing function f(D), f(0)=0.
Let:
$\displaystyle \phi(h(f(D))*h(f(D))*h'(f(D))*f'(D)=D$
where $\displaystyle \phi$ is the standard normal pdf
From the above equation, we can see that D spans the real line and the when the RHS is > 0 so is the LHS.
But, I can rewrite the above equation as:
$\displaystyle \frac{\partial\phi(h(f(D))}{\partial{D}}=D$
$\displaystyle \implies 0.5*D^2=\phi(h(f(D)) $
which does not make sense since the LHS is > 0 and the RHS is < 0