Coefficient Bounds for a Family of Analytic Functions Linked with a Petal-Shaped Domain and Applications to Borel Distribution

In this paper, by employing  sine hyperbolic inverse functions,  we  introduced the generalized  subfamily $\mathcal{RK}_{\sinh}(\beta)$ of analytic functions defined on the open unit disk $\Delta:=\{\xi: \xi \in \mathbb{C} \text{ and } |\xi|<1 \}$ associated with the petal-shaped domain. The bounds of the first three Taylor-Maclaurin's coefficients, Fekete-Szeg\"{o} functional and the second Hankel determinants are investigated for $f\in\mathcal{RK}_{\sinh}(\beta)$. We considered Borel distribution as an application to our main results. Consequently, a number of corollaries have been made based on our results, generalizing previous studies in this direction.