After this question : Dominated convergence 2.0?

I want to know, what about the case when $h\in L^1([0,1])$.

The completed question :

Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$. Assume that:

$\forall n\in\mathbb N, f_n''<h$, where $h \in L^1([0,1])$.

Is it true that $\lim \int_0^1 f_n=\int_0^1 g$ ?

After this question : Dominated convergence 2.0?

I want to know, what about the case when $h\in L^1([0,1])$.

The completed question :

Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$ and $\forall x \in [0,1], g(x)\in \mathbb R$. 

Assume that:

$\forall n\in\mathbb N, f_n''<h$, where $h \in L^1([0,1])$.

Is it true that $\lim \int_0^1 f_n=\int_0^1 g$ ?