A short proof of a theorem of Bertilsson by direct use of Loewner's method (Q1416628)

Let \(f(z)\) be an univalent function in the unit disk and let \(c_N((f')^p)\) be the \(N\)th Taylor coefficient of the function \((f'(z))^p\) for some \(p<0.\) In this paper the author uses Loewner's method for the inverse of an univalent function to give an elegant proof of the estimate \(| c_N((f')^p)| \leq| c_N((k')^p)| \) when \(1\leq N\leq 2| p| +1.\) Here \(k(z)=z/(1-z)^2\) is the Koebe's function. This result was obtained by \textit{D. Bertilsson} [Ark. Mat. 36, 255--273 (1998; Zbl 1025.30013)] with much more involved proof.