International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 38, Pages 2389-2400

Techniques of the differential subordination for domains bounded by conic sections

Stanisława Kanas

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.


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))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.