A note on neighborhoods of analytic functions having positive real part

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′.