Differential Evolution and a Conjecture on the Hankel Determinant in the Class of Starlike Functions

This paper continues our earlier work, in which the differential evolution algorithm was applied to coefficient problems in Geometric Function Theory, and the sharp $1/8$ bound of the fourth Hankel determinant $\det H_{4,1}\left(f\right)$ was identified numerically in the class ${\mathcal{S}}^{*}$ of starlike functions, i.e., of analytic univalent functions f in the unit disk $\mathbb{D}$ normalized by $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$, for which $f\left(\mathbb{D}\right)$ is starlike with respect to the origin. Here, the approach is extended to the fifth and sixth Hankel determinants $\det H_{5,1}\left(f\right)$ and $\det H_{6,1}\left(f\right)$. Using finite-dimensional reduction in the extremal problem provided by the Hummel–Sakaguchi theorem on support points of the Carathéodory class, we perform a fully documented and reproducible differential evolution search. In all independent runs, the values found agree, to within ${10}^{-14}$, with $16/625$ and $1/243$, respectively, and all maximizers found coincide, up to rotation, with the q-fold symmetric Koebe function $f_q\left(z\right)=z{(1-z^q)}^{-2/q}$, $q=5,6$. We prove that $|\det H_{q,1}\left(f_q\right)|={(2/q)}^{q-1}$ for every $q\in \mathbb{N}$ so that these values are attained in ${\mathcal{S}}^{*}$. Combining the numerical computations with the known sharp bounds for $q\le 3$, we conjecture that $|\det H_{q,1}{\left(f\right)|\le (2/q)}^{q-1}$ for all $f\in {\mathcal{S}}^{*}$ and $q=4,5,6$, with equality for $f_q$ and its rotations. In the whole class ${\mathcal{S}}^{*}$, these upper bounds are supported by numerical computations but are not proved; they are stated explicitly as conjectures. We also show that the pattern does not extend to all orders: For every $q\ge 10$, the function $f_{q-1}$ satisfies $|\det H_{q,1}\left(f_{q-1}\right)|=2^{q-2}(q-3)/{(q-1)}^q>{(2/q)}^{q-1}$ so that $f_q$ is not extremal for $q\ge 10$ and the general inequality $|\det H_{q,1}{\left(f\right)|\le (2/q)}^{q-1}$, $q\in \mathbb{N}$, is false.