Abstract
In this article, we establish new univalence criteria for normalized analytic functions f (z) = z + Sigma(infinity)(k=2) a(k) z(k )with f (z)/z not equal 0 in the unit disk U = {z : vertical bar z vertical bar < 1}. Indeed, we prove for any n >= 2 that the condition 1 vertical bar f (z)/z)((n))vertical bar <= (n!/(n + 1)) (1 - Sigma(n)(k=2)k vertical bar a(k)vertical bar ) is sufficient and sharp for f to be univalent in U. The equality attained for the functions f (z) = z + Sigma(n)(k=2)a(k) Z(k) , where Sigma(n)(k=2)k vertical bar a(k)vertical bar = 1. We investigate interesting geometric properties for such classes of functions. Nam ely, subordinations, inclusions, distortion and growth theorems, area estimate, starlikeness and convexity.