International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 38, Pages 2389-2400
doi:10.1155/S0161171203302212
Techniques of the differential subordination for domains bounded by conic sections
Department of Mathematics, Rzeszów University of Technology, W. Pola 2, Rzeszów Pl-35-959, Poland
Received 24 February 2003
Copyright © 2003 Stanisława Kanas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.