Verification of Brannan and Clunie's conjecture for certain subclasses of bi-univalent functions

Abstract

Let $\sigma $ denote the class of bi-univalent functions $f$, that is, both $f(z)=z+a_2z^2+\cdots $ and its inverse $f^{-1}$ are analytic and univalent on the unit disk. We consider the classes of strongly bi-close-to-convex functions of order $\alpha $ a