International Journal of Mathematics and Mathematical Sciences
Volume 24 (2000), Issue 8, Pages 563-568
doi:10.1155/S016117120000140X
Subordination by convex functions
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.
Abstract
Let K(α), 0≤α<1, denote the class of functions g(z)=z+Σn=2∞anzn 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 f∈K; and (ii) f(z)<g(z) at least in |z|<r1,r1((51−242)/23)1/2=0.8612… if
g∈K(1/2).