Second Hankel determinant for a subclass of analytic functions defined by S$\check{a}$l$\check{a}$gean-difference operator

Abstract

In the present investigation, inspired by the work on Yamaguchi type class of analytic functions satisfyingthe analytic criteria $\mathfrak{Re}\{\frac{f (z)}{z}\} > 0, $ in the openunit disk $\Delta=\{z \in \mathbb{C}\colon |z|<1\}$ and making use of S\v{a}l\v{a}gean-difference operator, which is a special type of Dunkl operator with Dunkl constant $\vartheta$ in $\Delta$ , wedesignate definite new classes of analytic functions $\mathcal{R}_{\lambda}^{\beta}(\psi)$ in $\Delta$. For functionsin this new class , significantcoefficient estimates $|a_2|$ and $a_3|$ are obtained. Moreover, Fekete-Szeg\H{o} inequalities and second Hankel determinant for the function belonging to this class are derived. By fixing the parameters a number of special cases are developed are new (or generalization) of the results of earlier researchers in this direction.