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In this study, we introduce and investigate two new subclasses of the bi-univalent functions; both f (z) and f −1 (z) are m-fold symmetric analytic functions. Among other results, upper bounds for the coefficients |am+1| and |a2m+1| are found in this investigation.

We obtain an interesting subordination relation for Salagean-type certain analytic functions by using subordination theorem.

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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.

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