International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 8, Pages 563-568

Subordination by convex functions

Ram Singh and Sukhjit Singh

Department of Mathematics, Punjabi University, Patiala 147002, (Punjab), India

Received 18 January 1990

Copyright © 2000 Ram Singh and Sukhjit Singh. 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.


Let K(α), 0α<1, denote the class of functions g(z)=z+Σn=2anzn which are regular and univalently convex of order α in the unit disc U. Pursuing the problem initiated by Robinson in the present paper, among other things, we prove that if f is regular in U,f(0)=0, and f(z)+zf(z)<g(z)+zg(z) in U, then (i) f(z)<g(z) at least in |z|<r0,r0=5/3=0.745 if fK; and (ii) f(z)<g(z) at least in |z|<r1,r1((51242)/23)1/2=0.8612 if gK(1/2).