Fekete–Szegö functional problems for some subclasses of bi-univalent functions defined by Frasin differential operator

Abstract

In this paper, we introduce a new subclass $\mathcal {B}_{\Sigma }^{\zeta }(m,\gamma ,\lambda ;\varphi )$ of bi-univalent functions defined by a new differential operator of analytic functions involving binomial series due to Frasin (Bol Soc Paran Mat (in press), 2019) in the open unit disk. We obtain coefficient bounds for the Taylor–Maclaurin coefficients $|a_{2}|$ and $|a_{3}|$ of the function $f\in \mathcal {B}_{\Sigma }^{\zeta }(m,\gamma ,\lambda ;\varphi )$. Furthermore, we solve the Fekete–Szegö functional problem for functions in $\mathcal {B}_{\Sigma }^{\zeta }(m,\gamma ,\lambda ;\varphi )$. The results presented in this paper improve or generalize the earlier results of Peng and Han (Acta Math Sci 34(1):228–240, 2014) and Tang et al. (J Math Inequal 10(4):1063–1092, 2016).