Third Hankel Determinant and Zalcman Functional for Sakaguchi Type Starlike Functions Involving q-Derivative Operator Related with Sine Function

The purpose of this paper is to consider coefficient estimates for q-starlike function with respect to symmetric points associated with sine function \(\mathcal{SS}^*_ q(1+sin(z))\) consisting of analytic functions \(f\) normalized by \(f(0)=f'(0)-1=0\) in the open unit disk \(\mathcal{U}_d=\{ z:z\in \mathbb{C}\quad \text{and}\quad \left\vert z\right\vert <1\}\) satisfying the condition \(\dfrac{2[zD_qf(z)]}{f(z)-f(-z)}\prec{1+sin(z)}=\psi(z)\), for all \(z\in\mathcal{U}_d\) to derive certain coefficient estimates \(b_2,b_3\) etc and Fekete-Szeg"{o} inequality for \(f\in\mathcal{SS}^*_q(1+sin(z)).\) Further to investigate the possible upper bound of third order Hankel determinant and also the Zalcman functional for \(f\in\mathcal{SS}^*_q(1+sin(z))\).