Suppose that f:G→C is an analytic function with some properties in a region G. Theorem 1 shows that (G

f) is a smooth unlimited covering manifold of f(G). Theorem 2 gives the sufficient and necessary conditions under which f is univalent in a simply connected region. In theorem 3

an application of theorem 2 is given to a function f which is analytic in the unit disk B(0;1) and is restricted by the condition f(0)=f’(0)-1=0. The relations between the univalence of f and the coefficients of its Taylor expansion as well as its boundary values are discussed.