Consider the following complex function:

$ frac {1} {1+ lambda f (z) f ( frac {1} {z})}, $

or $ lambda $ is a real constant number and $ z $ is a complex variable.

I need to find a function $ G (z) $ such as

$ G (z) times G ( frac {1} {z}) = frac {1} {1+ lambda f (z) f ( frac {1} {z})}. $

Can we find $ G (z) $ as a function of $ f $?

Can we solve the problem if $ f $ is a polynomial function?