On a Briot-Bouquet Differential Subordination

Let p(z) be analytic in the unit disc Δ, let h(z) be convex (univalent) in Δ, and let β and γ be complex numbers. The authors show that if p(z) = zp′(z)(βp(z) + γ)−1 ≺ h(z) (where ≺ denotes subordination), then p(z) ≺ h(z). They prove, further, that if, in addition, the differential equation q(z) + zq′(z)(βq(z) + γ)−1 = h(z) has a univalent solution q(z), then the sharp subordination p(z) ≺ q(z) holds. Applications of these results in the field of univalent functions are given.