Derivatives of the maximum modulus of a starlike function (Q1313673)

Let \(f\) be analytic in the unit disk \(D\) and \(M(r,f)=\max\{| f(z)|\mid| z|= r\}\) for \(r\in (0,1)\). In this paper the author describes the very delicate construction of a function \(f\) and a sequence \((r_ n)_{n\in\mathbb{N}}= (r_ n(f))_{n\in\mathbb{N}}\) with the following properties: 1) \(f(0)= f'(0)- 1=0\), \(f\) is starlike in \(D\), 2) \(r_ n\in (0,1)\), \(\lim_{n\to\infty} r_ n=1\), for any \(n\in\mathbb{N}\), \({d^ 2\over dr^ 2} M(r_ n,f)= M''(r_ n,f)\) exists and \[ \lim_{n\to\infty} {(1- r_ n)^ 2 M''(r_ n,f)\over M(r,f)}= \infty. \] Compare: 1) \textit{T. Sheil-Small} [Proc. Lond. Math. Soc., III. Ser. 21, 577-613 (1970; Zbl 0205.383)]; 2) the author [Bull. Lond. Math. Soc. 13, 207-213 (1981; Zbl 0448.30014)].