Volume 58, pp. 568-581, 2023.

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

Piotr Jastrzȩbski and Adam Lecko


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 $f(0)=f'(0)-1=0$ such that $f(\Bbb D)$ 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.

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Key words

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

AMS subject classifications

65K05, 30C45, 30C50

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