On the fourth and fifth coefficients of strongly starlike functions (Q1921413)

The class \(S^*(\alpha)\), \(0< \alpha\leq 1\) of univalent \(\alpha\)-strongly starlike functions introduced by Brannan and Kirwan and Stankiewicz was defined as a set of all functions \(f(z)= z+ a_2 z^2+ a_3 z^3+\cdots\) regular in the unit disc \(U= \{z: |z|< 1\}\) such that \[ \Biggl|\arg {zf'(z)\over f(z)}\Biggr|< {\pi\alpha\over 2},\quad z\in U. \] In this paper, the authors give the sharp estimations of \(|a_4|\): \[ \begin{aligned} |a_4|&\leq {2\alpha\over 3}\quad\text{for } 0< \alpha\leq \sqrt{2/17},\\ |a_4|& \leq {2\alpha\over 9} (1+ 17\alpha^2)\quad\text{for } \sqrt{2/17}\leq \alpha\leq 1;\end{aligned} \] and for some \(\alpha\in (0; 0, 35016)\cup (0, 44624; 1)\) the estimations of \(|a_5|\): \[ \begin{aligned} |a_5|& \leq {\alpha\over 2}\quad \text{for } 228\alpha^4- 19\alpha^3+ 2\alpha^2+ 39\alpha- 9\leq 0,\\ |a_5|& \leq {\alpha^2\over 9} (7+ 38\alpha^2)\quad \text{for } 76\alpha^3- 60\alpha^2+ 32\alpha- 0\geq 0.\end{aligned} \] The extremal functions are such that \[ {zf'(z)\over f(z)}= \Biggl({1+ \varepsilon z^k\over 1- \varepsilon z^k}\Biggr)^\alpha, \] where the parameters \(\varepsilon\) and \(k\) are choosing separately for every case. The problem is still open for \(n= 5\), \(\alpha\in (0,35016; 0, 44624)\) and for \(n\geq 6\).