  The theory of slice regular functions of a quaternionic variable extends the
notion of holomorphic function to the quaternionic setting. This theory,
already rich of results, is sometimes surprisingly different from the theory of
holomorphic functions of a complex variable. However, several fundamental
results in the two environments are similar, even if their proofs for the case
of quaternions need new technical tools. In this paper we prove the
Landau-Toeplitz Theorem for slice regular functions, in a formulation that
involves an appropriate notion of regular $2$-diameter. We then show that the
Landau-Toeplitz inequalities hold in the case of the regular $n$-diameter, for
all $n\geq 2$. Finally, a $3$-diameter version of the Landau-Toeplitz Theorem
is proved using the notion of slice $3$-diameter.
