The logarithmic coefficients are very essential in the problems of univalent functions theory. The importance of the logarithmic coefficients is due to the fact that the bounds on logarithmic coefficients of f can transfer to the Taylor coefficients of univalent functions themselves or to their powers, via the Lebedev–Milin inequalities; therefore, it is interesting to investigate the Hankel determinant whose entries are logarithmic coefficients. The main purpose of this paper is to obtain the sharp bounds for the second Hankel determinant of logarithmic coefficients of strongly starlike functions and strongly convex functions.

In this paper, defining new interesting classes, λ-pseudo bi-starlike functions with respect to symmetrical points and λ-pseudo bi-convex functions with respect to symmetrical points in the open unit disk U, we obtain upper bounds for the initial coefficients of functions belonging to these new classes.

In this paper, we first obtained some initial logarithmic coefficient bounds on a subclass of bounded turning functions $\mathcal{R}_{\mathcal{L}}$ related to a lune domain. For functions belonging to this class, we determined the sharp bounds for the second Hankel determinant of logarithmic coefficients $H_{2,1}\left( F_{f}/2\right) $ of bounded turning functions related to a lune domain. Finally, we calculated the bounds of third Hankel determinant of logarithmic coefficients $H_{3,1}\left( F_{f}/2\right)$ of bounded turning functions associated with a lune domain.