Some Symmetric Holomorphic Function Subfamilies and Their Hankel Determinants Involving the $( \mathfrak{p} , \mathfrak{q} )$-Derivative Operator

Abstract

Coefficient function estimates play a vital role in building a mathematical framework for nonlocal problems, providing the groundwork necessary to develop innovative methods that enhance the precision and efficiency of engineering applications in applied science fields. In this study, we establish coefficient bounds for normalized holomorphic functions of the form
$$ \mathfrak{f} (z) = z + \sum_{ \ell =1}^\infty \mathfrak{b}_ { \jmath \ell +1} z^{ \jmath \ell +1}, \; \left(z \in \mathbb{U} := \{z \in \mathbb{C}:\,|z| < 1\}, \; \jmath \in \mathbb{N}:=\{1,2,3, \cdots\}, \; \mathfrak{b}_ { \jmath \ell +1} \in \mathbb{C}\right),$$
that belongs to some subfamilies of $ \jmath $-fold symmetric functions defined by the $( \mathfrak{p} , \mathfrak{q} )$-derivative operator. The derived estimates pertain to the bounds of $\left| \mathfrak{b}_ { \jmath +1}\right|$, $\left| \mathfrak{b}_ {2 \jmath +1}\right|$, and $\left| \mathfrak{b}_ {3 \jmath +1}\right|$, along with the Fekete--Szeg\"{o} funtional $\left| \mathfrak{b}_ {2 \jmath +1} -\mu \mathfrak{b}_{ \jmath +1}^2\right|$. Additionally, we obtain the upper bound of the second Hankel determinant $\left| \mathfrak{b}_ { \jmath +1} \mathfrak{b}_{3 \jmath +1} - \mathfrak{b}_{2 \jmath +1}^2\right|$, which serves as an important indicator of the relationship between the coefficients. Python 3.12 (2023) was used for graphical illustrations and verification to confirm the accuracy of our comprehensive theoretical results.