please see here(p.174-175 Elementary real and complex analysis By Georgiĭ Evgenʹevich Shilov):
Image snapshot from google books:
Question is, why is $\displaystyle |H(z)| \lt 1/2$ true?
please see here(p.174-175 Elementary real and complex analysis By Georgiĭ Evgenʹevich Shilov):
Image snapshot from google books:
Question is, why is $\displaystyle |H(z)| \lt 1/2$ true?
It is the definition of continuity in $0$. For any $\epsilon$ you can find $\delta$ such as $z$ lies in the disc centered in $0$ with radius $\delta$ implies $H(z)$ is in the disc centered on $H(0)=0$ with radius $\epsilon=1/2$.
It follows from continuity of polynomials: If $H$ is continuus at $z_0$ then for any $\varepsilon>0$ there exists $r>0$ such that if $|z-z_0|