  The purpose of this paper is to consider coefficient estimates in a class of
functions $\mathfrak{G}_{\vartheta}^{\kappa}(\mathcal{X},\varkappa)$ consisting
of analytic functions $f$ normalized by $f(0)=f'(0)-1=0$\ in the open unit disk
$\Delta=\{ z:z\in \mathbb{C}\quad \text{and}\quad \left\vert z\right\vert <1\}$
subordinating generalized telephone numbers, to derive certain coefficient
estimates $a_2,a_3$ and Fekete-Szeg\"{o} inequality for
$f\in\mathfrak{G}_{\vartheta}^{\kappa}(\mathcal{X},\varkappa)$. A similar
results have been done for the function $ f^{-1} $ and
$\log\dfrac{f(z)}{z}.$Similarly application of our results to certain functions
defined by using convolution products with a normalized analytic function is
given, and in particular we state Fekete-Szeg"{o} inequalities for subclasses
described through Poisson Borel and Pascal distribution series.
