ADDITIVE AUTOMORPHIC-FUNCTIONS AND BLOCH FUNCTIONS

A function f analytic in the unit disk D is said to be strongly unifor mly continuous hyperbolically, or SUCH, on a set E subset-of D if for each epsilon > 0 there exists a delta > 0 such that \f(z) - f(z')\ < e whenever z and z' are points in E and the hyperbolic distance between z and z' is less than delta. We show that f is a Bloch function in D if and only if Absolute value of f is SUCH in D. A function f is said to be additive automorphic in D relative to a Fuchsian group GAMMA if, for each gamma is-an-element-of GAMMA, there exists a constant A(gamm a) such that f(gamma(z)) = f(z) + A(gamma). We show that if an analyti c function f is additive automorphic in D relative to a Fuchsian group GAMMA, where GAMMA is either finitely generated or if the fundamental region F of GAMMA has the right kind of structure, and if Absolute va lue of f is SUCH in F, then f is a Bloch function. We show by example that some restrictions on GAMMA are needed.