Estimates on the initial coefficients are obtained for normalized analytic functions f in the open unit disk with f and its inverse g = f −1 satisfying the conditions that zf ′ (z)/f (z) and zg ′ (z)/g(z) are both subordinate to a starlike univalent function whose range is symmetric with respect to the real axis. Several related classes of functions are also considered, and connections to earlier known results are made.

The estimates for the second Hankel determinant a 2 a 4 -a 2 3 of the analytic function f (z) = z + a 2 z 2 + a 3 z 3 + · · · , for which either zf (z)/f (z) or 1 + zf (z)/f (z) is subordinate to a certain analytic function, are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike and lemniscate starlike functions are obtained.

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An analytic functionfdefined on the open unit disk is biunivalent if the functionfand its inversef-1are univalent in𝔻. Estimates for the initial coefficients of biunivalent functionsfare investigated whenfandf-1, respectively, belong to some subclasses of univalent functions. Some earlier results are shown to be special cases of our results.

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