Criteria for univalent functions in the unit disk

摘要

Let U denote the set of all normalized analytic functions f in the unit disk {pipe}z{pipe} <1 satisfying the condition It is well-known that U is contained in the class S of univalent analytic functions in D. In this paper we first show that the condition where f is a normalized analytic function in {pipe}z{pipe} <1 and where for n ∈ {3,4,...}, implies that f ∈ U. This result improves an earlier known result, with an alternate and simple proof. Next we show that if g,h ∈ S, then the function F_r defined by F＿r(z) = r-1F(rz), where F(z) = g(z)h(z) / z, belongs to U (and hence is univalent in {pipe}z{pipe} <1) whenever r ∈ (0, r_0], where r_0 ≈ 0. 3263. This value is close to 1/3, which is the conjectured value for this problem.