On the geometry of the coefficient body \((a_ 2,a_ 3)\) for bounded univalent functions and some related coefficient problems (Q1082476)

Let S(b), \(0<b\leq 1\), denote the class of bounded normalized univalent functions: \[ S(b)=\{f| \quad f(z)=b(z+a_ 2z^ 2+...),\quad | z| <1,\quad | f(z)| <1\}. \] The corresponding class with all the coefficients real is denoted by \(S_ R(b)\). \textit{L. Pietrasik} maximizes the product \(a_ 2a_ 3\) for \(S_ R(b)\)- functions by using the variational method [Estimation of the functional \(A_ 2A_ 3\) in the class of bounded symmetric univalent functions, Acta Univ. Łódź., Folia Math., II. Ser. (to appear)]. In this paper, among other things, sharp estimations of the functionals \(| a_ 2a_ 3|\), \(| a^ 3_ 2a_ 3|\) in the class S(b) are obtained. It seems interesting that the estimation of the functional \(| a_ 2a_ 3|\) in the class S(b) is analogous as in \(S_ R(b)\).