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 \(f(z)=\alpha z+a_{2}z^{2}+ a_{3}z^{3}+\cdots\) which maximize Rea 3.
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Dedicated to the memory of Walter Hengartner
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Barnard, R.W., Pearce, K. & Williams, G.B. Three Extremal Problems for Hyperbolically Convex Functions. Comput. Methods Funct. Theory 4, 97–109 (2004). https://doi.org/10.1007/BF03321058
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DOI: https://doi.org/10.1007/BF03321058