International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 2, Pages 109-117
doi:10.1155/S0161171200001666

Differential subordinations for fractional-linear transformations

Yong Chan Kim,1 Adam Lecko,2 Jae Ho Choi,3 and Megumi Saigo3

1Department of Mathematics, Yeungnam University, Gyongsan 712-749, Korea
2Department of Mathematics, Technical University of Rzeszów, Rzeszów 35-959, Poland
3Department of Mathematics, Fukuoka University, Fukuoka 814-0180, Japan

Received 29 May 1998; Revised 29 June 1998

Copyright © 2000 Yong Chan Kim et al. 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 establish that the differential subordinations of the forms p(z)+γzp′(z)≺h(A1,B1;z) or p(z)+γzp′(z)/p(z)≺h(A2,B2;z) implies p(z)≺h(A,B;z), where γ≥0 and h(A,B;z)=(1+Az)/(1+Bz) with −1≤B<A.