Notes on the norm of pre-Schwarzian derivatives on bi-univalent functions of order $\alpha$

Abstract

In the present paper we estimate the norm of the pre-Schwarzian derivative of bi-starlike functions of order $\alpha$ where $\alpha\in[0,1)$. Initially this problem was handled by Rahmatan et al. in [Bull Iran Math Soc {\bf43}: 1037-1043, 2017]. We pointed out that the proofs and bounds by Rahmatan et al. are incorrect and present correct proofs and bounds.