A function f analytic in the unit disk D is said to be strongly uniformly continuous hyperbolically, or SUCH, on a set E ⊂ D if for each ∊ > 0 there exists a δ > 0 such that |f(z) — f(z')| < ∊ whenever z and z' are points in E and the hyperbolic distance between z and z' is less than δ. We show that f is a Bloch function in D if and only if |f| is SUCH in D. A function f is said to be additive automorphic in D relative to a Fuchsian group F if, for each γ ∊ Γ, there exists a constant Aγ such that f(γ(z)) =f(z) + Aγ. We show that if an analytic function f is additive automorphic in D relative to a Fuchsian group Γ, where Γ is either finitely generated or if the fundamental region F of Γ has the right kind of structure, and if |f| is SUCH in F, then f is a Bloch function. We show by example that some restrictions on Γ are needed.