We study a family of inequalities formed by the Fekete-Szegö design, making use of the normalized analytic functions in the open unit disk. We investigate the following functional: \[\Psi(z):= \dfrac{z^{1-ǎrtheta}si'(z)}{si^{1-ǎrtheta}(z)},\] where $\vartheta \geq 0$ acts on a domain having the starlike with respect to the boundary of the unit disk and symmetric with respect to the real axis. In addition, various presentations of the central result for functions formulated by convolution are investigated. As a special instance of this result, Fekete-Szegö issue associated with Special functions (differential operators) is studied. Moreover, by using bounds of the initial Taylor coefficients, we discussed Second Hankel determinant results.