Sharp Coefficient Results on the Inverse of Silverman Starlike Functions

Abstract

In the present paper, we consider a subclass of starlike functions \(\mathcal{G}_{\mu}\) introduced by Silverman. It is defined by the ratio of analytic representations of convex and starlike functions. The main aim is to determine the sharp bounds of coefficient problems for the inverse of functions in this class. We derive the upper bounds of some initial coefficients, the Fekete–Szegö type inequality and the second Hankel determinant \(\mathcal{H}_{2,2}\left(f^{-1}\right)\) for \(f\in\mathcal{G}_{\mu}\). On the third Hankel determinant \(\mathcal{H}_{3,1}\left(f^{-1}\right)\), we give a bound on the inverse of \(f\in\mathcal{G}\). All the results are proved to be sharp.