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. © 2013 Austral Internet Publishing. All rights reserved.