Let D be a domain. Let {fn} be a set of functions that are holomorphic on D and is normal. I’m trying to prove that {fn’} is also a normal family.
The following is my attempt:
Note that if {Gn} is uniformly convergent on every compact subset of D, then so is {Gn’}.
Now, given any subset of {fn’}.
These indices correspond to a subsequence {fn,k} in {fn}. Since this family is normal, there exists a subsequence that is uniformly convergent on every compact subset of D, which is denoted by {Fn,k}.
Hence this {Fn,k} again corresponds to a subsequence of the original sequence {fn’｝and this subsequence is uniformly convergent on every compact subset of D, showing that {fn’} is normal.
But I am not sure whether this attempt works. Can someone tell me if my attempt is correct?
And I think this can be proved by using Cauchy Integral formula and some estimation.
Could someone use this idea to prove?