Techniques of the differential subordination for domains bounded by conic sections

Abstract

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.