Suppose that f is an analytic function in the region D which contains the point a. Let
\[ F ( z ) = z - a - q f ( z ) , \quad { \mathrm { w h e r e } } \quad q { \mathrm { ~ i s ~ a ~ c o m p l e x ~ p a r a m e t e r . } } \]
Let \(K \subset D\) be a circle with the center at point a and also we assume that \(f ( z ) \neq 0\) for \(z \in K\) . Prove that the function F has one and only one zero \(z = w\) on the closed disc K whose boundary is the circle K if \(| q | < \operatorname* { m i n } _ { z \in K } { \frac { | z - a | } { | f ( z ) | } }\)
Let \(G ( z )\) be an analytic function on the disk K. Apply the residue theorem to prove that \(\frac { G ( w ) } { F ^ { \prime } ( w ) } = \frac { 1 } { 2 \pi i } \int _ { K } \frac { G ( z ) } { F ( z ) }\) dz, where w is the zero from (1).
\(\mathrm { I f } ~ z \in { K }\) , prove that the function \(\displaystyle \frac { 1 } { F ( z ) }\) can be represented as a convergent series with respect to \(q \colon { \frac { 1 } { F ( z ) } } = \sum _ { n = 0 } ^ { \infty } { \frac { ( q f ( z ) ) ^ { n } } { ( z - a ) ^ { n + 1 } } }\)