Expectation closure error for a nonlinear recurrence

A bound $\mathbb{E}[B_{j+1}\mid B_j]\le f(B_j)$ implies $\mathbb{E}{B}_{j+1}\le \mathbb{E}f(B_j)$. It does not by itself imply $\mathbb{E}{B}_{j+1}\le f(\mathbb{E}{B}_j)$. For $f(b)=b^2/n$, Jensen's inequality goes in the opposite direction. The missing contribution is the variance: $\mathbb{E}[B_j^2]=(\mathbb{E}{B}_j{)}^2+\mathrm{Var}(B_j)$.