Briot-Bouquet Differential Subordination and Bernardi's Integral Operator

The conditions on A, B, beta and gamma are obtained for an analytic function p defined on the open unit disc D and normalized by p(0) = 1 to be subordinate to (1 + Az)/(1 + Bz), -1 <= B < A <= 1 when p(z) + zp '(z)/(beta p(z) + gamma) is subordinate to e(z). The conditions on these parameters are derived for the function p to be subordinate to root 1 + z or e(z) when p(z) + zp '(z)/(beta p(z) + gamma) is subordinate to (1 + Az)/(1 + Bz). The conditions on beta and gamma are determined for the function p to be subordinate to e(z) when p(z)+zp '(z)/(beta p(z)+gamma) is subordinate to root 1 + z. Related result for the function p(z)+zp '(z)/(beta p(z)+ gamma) to be in the parabolic region bounded by the Rew = |w - 1| is investigated. Sufficient conditions for the Bernardi's integral operator to belong to the various subclasses of starlike functions are obtained as applications.