Uniformly convex functions

Recently, A. W. Goodman introduced the geometrically defined class UCV of uniformly convex functions on the unit disk; he established some theorems and raised a number of interesting open problems for this class. We give a number of new results for this class. Our main theorem is a new characterization for the class UCV which enables us to obtain subordination results for the family. These subordination results immediately yield sharp growth, distortion, rotation and covering theorems plus sharp bounds on the second and third coefficients. We exhibit a function k in UCV which, up to rotation, is the sole extremal function for these problems. However, we show that this function cannot be extremal for the sharp upper bound on the nth coefficient for all n. We establish this by obtaining the correct order of growth for the sharp upper bound on the nth coefficient over the class UCV and then demonstrating that the nth coefficient of k has a smaller order of growth.