A note on neighborhoods of analytic functions having positive real part (Q916822)

The author proves: If \(p(z)=1+\sum^{\infty}_{k=1}p_ kz^ k\) is holomorphic in the unit disk D and \(Re(zp(z))'>0\) for \(z\in D\) and if \(\sum^{\infty}_{k=1}| p_ k-q_ k| \leq 2 \ln 2-1\) then \(q(z)=1+\sum^{\infty}_{k=1}q_ kz^ k\) is holomorphic in D and \(Re(q(z))>0\) for \(z\in D\). The result is sharp.