Approximation of a function f on $\mathbb{T}$

First of all, I apologize if this has been asked before, but I could not find it.

Let $f : \mathbb{T} \to \mathbb{C}$ be an $n$-times differentiable function such $f^{(n)}$ is bounded. I must specify that we talk about differentiability everywhere on the circle $\mathbb{T}$, and, this is important, with respect to the complex variable (I do not only consider the $2\pi$-periodic function associated to $f$).

My question is the following: does there exist a sequence $(f_p)_p$ of elements of $C^n(\mathbb{T})$ such that :

1. $f_p^{(j)}$ converges to $f^{(j)}$ uniformly on $\mathbb{T}$ for $1\leq j \leq n-1$.
2. $f_p^{(n)}$ is pointwise convergent to $f^{(n)}$ on $\mathbb{T}$ (everywhere !).
3. $|f_p^{(n)}(z)| \leq C \|f^{(n)} \|_{\infty}$ (say with $C=2$ for example).

Of course, one could think of setting $f_j = f \ast F_j$ where $F_j$ is the Fejer kernel. It works well for $1.$ and $3.$, but it is not clear at all that $2.$ will hold true.