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), zp′(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) + (zp′(z)/(βp(z) + γ))≺pk(z), when the function pk(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.