In 5 pages, please try to prove Theorem 3 based on Montel''s Theorem. please use "Latex" Knuth Donald to write this paper.
It is known that Theorem 3 on page 137 of the attached Krantz''s book would make a very nice "central result". In order to prove it, one needs in particular to have a good understanding of the topology of uniform convergence. As part of this, it is necessary to revise the detail of Montel''s Theorem (chapter one), since uniform convergence is intimately related to normal families of holomorphic functions. But another very important issue, not explicitly discussed by Krantz, is how the topology of uniform convergence relies on the assumption that any complex domain we are considering is a complete metric space.
Attached is
1- A plan for my thesis attached
2- Introduction
3- Some information about Theorem 3 with an important question that need to answered.
4- Krantz''s book, the main and important source for the thesis.
I need to keep up with an expert to finalise part of the thesis each week.