Briot–Bouquet differential superordinations and sandwich theorems

Abstract

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 ) + z p ′ ( z ) β p ( 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 Ω 1 and Ω 2 in the complex plane the authors determine properties of functions p satisfying a Briot–Bouquet sandwich of the form Ω 1 ⊂ { p ( z ) + z p ′ ( z ) β p ( z ) + γ | z ∈ U } ⊂ Ω 2 . Generalizations of this result are also considered.