It is known that if a sequence of continously differentiable, convex functions $f_n: [a,b] \longrightarrow [c,d]$ converges uniformly to a continously differentiable function $f: [a,b] \longrightarrow [c,d]$, that the derivatives $f_n^\prime$ converges pointwise to $f^\prime$. But the convergence doesn't need to be uniformly, see
Convex functions and uniform convergence of derivatives
Does the convergence of $f_n^\prime$ to $f^\prime$ hold in the $L^1$-sense?