[For part I see the the authors in J. Comput. Appl. Math. 105, No. 1-2, 327-336 (1999; Zbl 0944.30008).]
Summary: In the previous paper, due to the authors, the class of -uniformly convex functions, , has been introduced. The mentioned class, denoted by -, is a generalization of the class of uniformly convex functions introduced by Goodman, and studied by Rønning, Ma and Minda. This paper is a continuation of the investigation of the class -, for that the region of values of , where -, is a domain bounded by the conic curves, which kind depends of the parameter . The estimates of , bounds of coefficients of -uniformly convex functions, the sharp bounds on , for , and the extremal functions which realize equality are given. Some particular examples of functions having the required properties are found.