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The concept of differential subordination was introduced in [4] by S. S. Miller and P. T. Mocanu and the concept of strong differential subordination was introduced in [1] by J. A. Antonino and S. Romaguera. This last concept was applied in the special case of Briot-Bouquet strong differential subordination. In this paper we study the strong differential subordinations in the general case, following the general theory of differential subordinations presented in [4].

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The authors introduce new classes of analytic functions in the open unit disc which are defined by using multiplier transformations. The properties of these classes will be studied by using techniques involving the Briot-Bouquet differential subordinations. Also an integral transform is established.

Let p be a complex-valued harmonic function in the unit disc U of the form p(z) = p1(z) + p2(z), where p1 and p2 are analytic in U. In [5] the authors have determined properties of the function p such that p satisfies the differential subordination ϕ(p(z), Dp(z), D 2 p(z); z) ⊂ Ω ⇒ p(U) ⊂ ∆. In this article, we consider the dual problem of determining properties of the function p, such that p satisfies the second-order differential superordination Ω ⊂ ϕ(p(z), Dp(z), D 2 p(z); z) ⇒ ∆ ⊂ p(U).