 Convex-payoff delta monotonicity in a binomial market

A convex terminal payoff has a convex pricing extension in the binomial market, because risk-neutral pricing is a positive weighted sum of rescaled copies of the payoff. Let the stock factors be $u,d$, the riskless gross factor be $R$, and $q=(R-d)/(u-d)$. The replicating portfolio in a binomial market satisfiesThe weights sum to one. Convexity orders the secant slopes on the adjacent intervals $[d^{2}s,uds]$ and $[uds,u^{2}s]$, proving the displayed inequalities. The inequalities need not be strict: an affine payoff has constant stock holding.