  Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with
$|w_0| < |z_0|$, and $\mathcal{H}_0$ be the class of all analytic self-maps $f$
of $\mathbb{D}$ normalized by $f(0)=0$. In this paper, we establish the
fourth-order Dieudonn\'e's Lemma and apply it to determine the variability
region $\{f^{(4)}(z_0): f\in \mathcal{H}_0,f(z_0) =w_0, f'(z_0)=w_1,
f''(z_0)=w_2\}$ for given $z_0,w_0,w_1,w_2$ and give the form of all the
extremal functions.
