Techniques of the differential subordination for domains bounded by conic sections

We solve the problem of finding the largest domain D for which, under given ψ and q , the differential subordination ψ ( p ( z ) , z p ′ ( z ) ) ∈ D ⇒ p ( z ) ≺ q ( z ) , where D and q ( 𝒰 ) are regions bounded by conic sections, is satisfied. The shape of the domain D is described by the shape of q ( 𝒰 ) . Also, we find the best dominant of the differential subordination p ( z ) + ( z p ′ ( z ) / ( β p ( z ) + γ ) ) ≺ p k ( z ) , when the function p k ( k ∈ [ 0 , ∞ ) ) maps the unit disk onto a conical domain contained in a right half-plane. Various applications in the theory of univalent functions are also given.