MATH 6322, Complex Analysis Fall 2011, HW#9 (all about normal family)
Due date: Wed., Nov. 30, 2011
Problem 1
. Page 204, #20, #24 on the textbook.
The following consists a small project about the normal family. You are
first asked to prove the Marty’s theorem. You are then asked to prove the
socalled
Zalcman’s theorem
.
The final goal is to prove a generalization of
Montel’s theorem (which corresponds to the little picard theorem).
Note
that the original Montel’s theorem we learnt corresponds to the Liouville’s
theorem (why?).
Recall the Montels theorem we have learnt:
a family of holomporphic
functions on a region of
C
is a normal family (in the sense that every sequence
of functions in the family admits a subsequence that converges uniformly on
compact sets to an analytic function) if the family is locally bounded, that
is, for every compact subset
K
there is a constant
C
K
such that

f
(
z
)
 ≤
C
K
for all
z
∈
K
and every
f
in the family.
Next, you are asked to prove Marty’s theorem, which gives a necessary
and sufficient condition for a family of holomrphic functions on region of
C
to be a normal family in the
extended sense
that every sequence of
functions in the family admits either a subsequence that converges uniformly
on compact sets to an analytic function
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 Summer '08
 Staff
 Math, Marty, Normal family, Little Picard Theorem, Marty’s theorem

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