A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions

We give a Fekete-Szeg? type inequality for an analytic function on the unit disk with Bloch seminorm ≤1. As an application of it, we derive a sharp inequality for the third coefficient of a uniformly locally univalent function f(z) = z + a ₂ z ² + a ₃ z ³ + ⋯ on the unit disk with pre-Schwarzian norm ≤λ for a given λ > 0. The first author was partially supported by the JSPS Grant-in-Aid for Scientific Research (B), 17340039. Authors’ addresses: T. Sugawa and T. Terada, Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan Current address: T. Sugawa, Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan