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.

Let S * e denote the class of analytic functions f in the open unit disk normalized by f (0) = f (0) − 1 = 0 and satisfying the condition zThe structural formula, inclusion relations, coefficient estimates, growth and distortion results, subordination theorems and various radii constants for functions in the class S * e are obtained. In addition, the sharp S * e -radii for functions belonging to several interesting classes are also determined.

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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