Expected Utility and Jensen's Inequality

Consider two random variables (costs and valuations) distributed $vbacksim G(.)$ and
$c backsim F(.)$ with pdfs $g(.)$ and $f(.)$. Let the supports of $c$ and
$v$ be $[x,y]$. Let $x<a=E(v)<b<y$, so $[a,b]subsetlbrack x,y]$. Now
consider a strictly concave (twice differentiable and continuous) utility function
$u(.)$, with $u^{prime}(.)>0$, $u^{^{primeprime}}(.)<0$, and $u(0)=0$
(passes through the origin). Establish sufficient conditions such that the
expression
$$int_{a}^{b}u(E(v)-c)f(c)dc-int_{a}^{b}int_{x}^{y}%
u(E(v)-v)g(v)f(c)dvdcgeq0,$$ where $E(v)=int_{x}^{y}vg(v)dv.$

Things I’ve tried:

1. $int_{0}^{bar{v}}u(E(v)-v)g(v)dvleq0$ by Jensen’s inequality. To see
   this, let $E(v)-v=t$. But $E(t)=E_{v}[E(v)-v]=0$, and so $E(u(t))leq
   u(E(t))=0$, since $u(0)=0$ by assumption.

2. Clearly, $int_{a}^{b}u(E(v)-c)f(c)dcleq0$, since we are integrating the
   integrand $(E(v)-c)$ from $a=E(v)$ to $b$.

3. Intuitively, a variant of Jensen’s inequality should apply if $c$ and $v$
   are i.i.d. Let $c$ and $v$ be i.i.d. with identical supports. Then the
   integrands are the same, and we have the expression
   $$int_{a}^{b}%
   u(E(v)-v)f(c)dc-int_{a}^{b}int_{x}^{y}u(E(v)-v)g(v)f(c)dvdc.$$
   However, we
   can’t apply Jensen’s inequality directly since $int_{a}^{b}u(E(v)-v)f(c)dc$
   is not $u(E(x))$, even if we “factor out” the outer integrals. $int_{x}%
   ^{y}u(.)g(v)dv$ seems to be a form of $E(u(x))$.

At a loss as to what to do here. Any help would be greatly appreciated. Thank you!