Computational Methods and Function Theory 4 (2004), No. 1, 97--109
Copyright Heldermann Verlag 2004
Roger W. Barnard
barnard@math.ttu.edu , Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, U.S.A.
Kent Pearce
pearce@math.ttu.edu , Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, U.S.A.
G. Brock Williams
williams@math.ttu.edu , Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, U.S.A.
In this paper we apply a variational method to three extremal problems for hyperbolically convex functions posed by W. Ma and D. Minda [Ann. Polon. Math. 60 (1994) 81--100] and Ch. Pommerenke [private communication]. 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.
Keywords: Hyperbolically convex functions, Julia variation.
MSC 2000: Primary 30C70.
[FullText-pdf (392 K)] [FullText-ps (412 K)]