On the Taylor coefficients for powers of the derivative of univalent functions (Q1282014)

Let \(S\) denote the class of all functions \(f\) analytic and univalent in the unit disc \(D:=\{z\in\mathbb{C}: | z| <1\}\) and such that \(f (0)=0\), \(f'(0)=1\). For every \(f\in S\) and every \(p\in\mathbb{R}\) the function \(z\to (f'(z))^p=\exp (p\log f'(z))\) with \((f'(0))^p=1\) is analytic in \(D\), i.e., has a power series expansion at \(z=0\): \((f'(z))^p= \sum^\infty_{n=0} c_{n,p}[f]z^n\), \(c_{0,p} [f]=1\). The author gives ``a necessary condition such that the Koebe function \(z/(1+z)^2\) in an extremal function for the problem of maximizing the modulus of the \(n\)th Taylor coefficient of the function \((f')^p\), \(p\in\mathbb{R}\). The proof uses an elementary version of Pontryagin's maximum principle applied to Löwner type differential equations. The author proves also that the Koebe function \(z/(1+z)^2\) is not the extremal function for the \(n\)th Taylor coefficient of the functions \(1/f'\) for all \(n\geq 6\)''.