For α>0, let K(k,α) denote the class of normalized k-convex functions defined on the unit disk D. Loosely speaking, this means that fis univalent and at each point of the boundary of f(D) the euclidean curvature is at least k. Note that K(0,1) is the classical family K of normalized convex functions. Tne families K(k,α) are closely related to convex functions of bounded type as introduced by Goodman. Our main result is that the family K(k,α) is mapped into K by composition with a certain linear fractional transformation- This fact enables us to give a unified treatment of a number of known results for the families K(k,α) and convex functions of bounded type and to establish sharp new results by making use of various theorems for the family K. There is an interesting phenomenon concerning extremal functions. There is a single extremal function, which is unique up to rotation, for the sharp upper and lower bounds on and the sharp lower bound on . On the other hand, for the upper bound on this same function is extremal only when the modulus of z not too close to one.