  We study martingale inequalities from an analytic point of view and show that
a general martingale inequality can be reduced to a pair of deterministic
inequalities in a small number of variables. More precisely, the optimal bound
in the martingale inequality is determined by a fixed point of a simple
nonlinear operator involving a concave envelope. Our results yield an
explanation for certain inequalities that arise in mathematical finance in the
context of robust hedging.
