On a conjecture of Fournier, Ma, and Ruscheweyh for bounded convex functions (Q648505)

Let \(C_b\) stand for the class of univalent functions that are bounded and convex on the unit disk. \textit{D. R. Fournier}, \textit{J. Ma} and \textit{S. Ruschweyh} [in: Approximation theory. In memory of A. K. Varma. New York, NY: Marcel Dekker. Pure Appl. Math., Marcel Dekker. 212, 225--241 (1998; Zbl 0912.30007)] have proved the inequality \(|a_4|\leq{7\over 3}|a_1|+ 2(3M)^{-1}\) when \(f(D)\subset\{w:|w|< M\}\). They conjectured, that the number \({7\over 3}\) may be replaced by 2 and that the constant is sharp for a function that maps \(D\) onto a triangular region symmetric w.r.t. the real axis. The present authors confine their considerations to a dense subclass \({\mathcal K}_M\subset C_M\) that consists of functions for which \(f(D)\) is bounded by either straight edges (proper sides) or arcs of \(|w|=M\). By making use of the Julia-type variational technique they are able to show the following theorem: The extremal value of the functional \([|a_4|- 2(3M)^{-1}]|a_2|^{-1}\) over the class \({\mathcal K}_M\), if it exists, can be obtained by a function that maps \(D\) onto a region with at most 3 proper sides. The authors examine some functions with this property and find a great deal of numerical evidence to suggest that 2 is actually the best bound over \(C_b\).