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 ifΩ⊂{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 Ω1Ω1 and Ω2Ω2 in the complex plane the authors determine properties of functions p satisfying a Briot–Bouquet sandwich of the formΩ1⊂{p(z)+zp′(z)βp(z)+γ|z∈U}⊂Ω2. Generalizations of this result are also considered.