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Techniques of the differential subordination for domains bounded by conic sections. (English) Zbl 1130.30307
Summary: 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))Dp(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))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.
MSC:
30C45Special classes of univalent and multivalent functions
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
33E05Elliptic functions and integrals
30C35General theory of conformal mappings