Volume 58, pp. 568-581, 2023.

An evolutionary approach to the coefficient problems in the class of starlike functions

Piotr Jastrzȩbski and Adam Lecko

Abstract

In this paper, we apply the differential evolution algorithm as a new approach to solve some coefficient problems within Geometric Function Theory. We find sharp bounds for the determinant of the Hankel matrix $H_{4,1}(f)$ and the determinants of all its sub-matrices for the class of starlike functions, i.e., for the class of all analytic injective functions $f$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ normalized by $f(0)=f^{\prime }(0)-1=0$ such that $f(\mathbb{D})$ is a starlike set with respect to the origin. In addition, a relevant conjecture regarding some coefficient functionals related to the Zalcman functional is formulated.

Full Text (PDF) [302 KB], BibTeX , DOI: 10.1553/etna_vol58s568

Key words

differential evolution algorithm, Hankel determinant, starlike function, Carathéodory class and Zalcman functional

AMS subject classifications

65K05, 30C45, 30C50