A coefficient product estimate for bounded univalent functions (Q2769236)

Let \(S(M)\), \(M>1\), denote the class of holomorphic and univalent functions \(f\) in the unit disc \(D=\{z\in \mathbb{C}:|z|< 1\}\) which have the form \(f(z)= z+ a_2z^2+ \cdots+ a_nz^n+ \dots\), \(z\in D\) and are bounded by \(M\) in \(D\). Let \(P_M\), \(P_M(0)=0\), denote the function given by the formula NEWLINE\[NEWLINEP_m(z)= 2z \left[(1-z)^2 +{1\over M}z+(1-z) \sqrt{(1-z)^2 +{4\over M}z}\right]^{-1}z\in D.NEWLINE\]NEWLINE It is known that, in the class \(S(M)\) the Pick function \(P_M\) is not extremal for \(\max |a_3|\) if \(M\in(1,e)\). However, for the functional \(|a_2a_n|\), \(n\geq 3\) the Pick function is extremal if \(M\) is close to 1 [see: \textit{Z. J. Jakubowski}, \textit{D. V. Prokhorov}, \textit{J. Szynal}, Complex Variables, Theory Appl. 42, No. 3, 241-258 (2000); \textit{Z. J. Jakubowski}, in: Univalent Functions, fractional calculus and their appl. 75-86 (1989; Zbl 0715.30014)]. For given integers \(m\) and \(n\), \(2<m<n\), the authors of this paper consider the problem \(\max|a_m a_n|\) in the class \(S(M)\). He proves that \(\max|a_ma_n |\) is realised by the Pick function for \(M\) close to 1 iff \((m-1)\) and \((n-1)\) are relatively prime. If \((m-1)\) and \((n-1)\) are not relatively prime then the functional \(|a_ma_n|\) is not maximized by rotations of the Pick function, when \(M\) is close to 1.