A note on neighborhoods of analytic functions having positive real part

• Main Author: Janice B. Walker
• Format: Article
• Language: English
• Published: Hindawi Limited 1990-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Subjects: functions having positive real part (Carathéodory class); subordinate function; δ-neighborhood; and convolution (Hadamard product).
• Online Access: http://dx.doi.org/10.1155/S0161171290000643

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′.