International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 3, Pages 425-429
doi:10.1155/S0161171290000643
A note on neighborhoods of analytic functions having positive real part
Department of Mathematics, Xavier University, Cincinnati 45207, Ohio, USA
Received 7 July 1989; Revised 18 October 1989
Copyright © 1990 Janice B. Walker. 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 P denote the set of all functions analytic in the unit disk D={z||z|<1} having the form p(z)=1+∑k=1∞pkzk with Re{p(z)}>0. For δ≥0, let Nδ(p) be those functions q(z)=1+∑k=1∞qkzk analytic in D with ∑k=1∞|pk−qk|≤δ. We denote by P′ the class of functions analytic in D having the form p(z)=1+∑k=1∞pkzk with Re{[zp(z)]′}>0. We show that P′ is a subclass of P and detemine δ so that Nδ(p)⊂P for p∈P′.