Three extremal problems for hyperbolically convex functions

Abstract. In this paper we apply a variational method to three extremal problems for hyperbolically convex functions posed by Ma and Minda and Pommerenke [6, 14]. We first consider the problem of extremizing Re f(z)/z. We determine the minimal value and give a new proof of the maximal value previously determined by Ma and Minda. We also describe the geometry of the hyperbolically convex functions f(z) = αz+a2z2 +a3z3 + · · · which maximize Re a3. 1