Criteria and geometric properties for bounded univalent functions in the unit disk

Abstract

In this article, we establish new univalence criteria for normalized analytic functions f(z) = z + \(\sum_{k=2}^{\infty}\) a_k z^k with f(z)/z ≠ 0 in the unit disk \(\mathbb{U}\) = {z : |z| < 1}.  Indeed, we prove for any n ≥ 2 that the condition |(f(z)/z)⁽ⁿ⁾| ≤ (n!/(n+1))(1 − \(\sum_{k=2}^{n}\)k|a_k|) is sufficient and sharp for f to be univalent in \(\mathbb{U}\).  The equality attained for the functions f(z) = z + \(\sum_{k=2}^{n}\) a_k z^k, where \(\sum_{k=2}^{n}\)k|a_k| = 1.  We investigate interesting geometric properties for such classes of functions.  Namely, subordinations, inclusions, distortion and growth theorems, area estimate, starlikeness and convexity.