Briot–Bouquet differential superordinations and sandwich theorems

Briot–Bouquet differential subordinations play a prominent role in the theory of differential subordinations. In this article we consider the dual problem of Briot–Bouquet differential superordinations. Let β and γ be complex numbers, and let Ω be any set in the complex plane C. The function p analytic in the unit disk U is said to be a solution of the Briot–Bouquet differential superordination if Ω ⊂ p(z)+ zp (z) βp(z)+ γ z ∈ U . The authors determine properties of functions p satisfying this differential superordination and also some generalized versions of it. In addition, for sets Ω1 and Ω2 in the complex plane the authors determine properties of functions p satisfying a Briot–Bouquet sandwich of the form Ω1 ⊂ p(z)+ zp (z) βp(z)+γ z ∈ U ⊂ Ω2. Generalizations of this result are also considered. © 2006 Elsevier Inc. All rights reserved.