If $(X_n)_{n \in \mathbb{N}}$ is a nonnegative submartingale for the filtration $(\mathcal{F}_n)_n$ and F is an increasing and convexe function. We suppose that F has a differentiable function f. (F and f are nonnegative)
Let $X'_{n}=\max_{0 \leq k \leq n}{X_k};Y_n=F(X'_n)-(X'_n-X_n)f(X'_n)$
Prove that $(Y_n)_n$ is a submartingale.
They gave a hint : prove that $Y_{n+1} - Y_n \geq f(X'_n)(X_{n+1}-X_n).$
We have $Y_{n+1}-Y_n=F(X'_{n+1})-(X'_{n+1}-X_{n+1})f(X'_{n+1})-F(X'_n)+(X'_n-X_n)f(X'_n)=F(X'_{n+1})-F(X'_n)+f(X'_n)(X'_n-X'_{n+1}+X_{n+1}-X_n)$
then how can we prove
$F(X'_{n+1})-F(X'_n)+f(X'_n)(X'_n-X'_{n+1}+X_{n+1}-X_n) \geq f(X'_n)(X_{n+1}-X_n)$