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

Abstract

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'(\zeta)^{\vartheta}\left( \frac{\zeta f'(\zeta )}{f(\zeta )}\right)^{1-\vartheta} \prec 1+\sin \zeta ; \qquad (0\leq \vartheta \leq 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.

Keywords

• analytic function
• starlike function
• subordination
• coefficient problem
• fekete-szegöinequality
• hankel determinant
• poisson distribution series