Fekete-Szegö and Hankel inequalities for certain class of analytic functions related to the sine function

In this present investigation, the authors obtain Fekete-Szegö inequality for certain normalized analytic function $f(\zeta )$ defined on the open unit disk for which

$$(f^{\prime }(\zeta {)}^{\vartheta }{\left(\frac{\zeta f^{\prime }(\zeta )}{f(\zeta )}\right)}^{1-\vartheta }\prec 1+\sin \zeta ;(0\le \vartheta \le 1)$$

lies in a region starlike with respect to $1$ and symmetric with respect to the real axis. As a special case of this result, the Fekete-Szegö inequality for a class of functions defined through Poisson distribution series is obtained. Further, we discuss the second Hankel inequality for functions in this new class.