Techniques of the differential subordination for domains bounded by conic sections

Abstract

We solve the problem of finding the largest domainDfor which, under givenψandq, the differential subordinationψ(p(z),zp′(z))∈D⇒p(z)≺q(z), whereDandq(𝒰)are regions bounded by conic sections, is satisfied. The shape of the domainDis described by the shape ofq(𝒰). Also, we find the best dominant of the differential subordinationp(z)+(zp′(z)/(βp(z)+γ))≺pk(z), when the functionpk(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.