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Exercise 1.
Let
be a domain, symmetric with respect to the real axis,
having a nonempty intersection
with it. Each holomorphic function in D can be expressed
uniquely in the form
for each
 where
g and h are two
holomorphic functions in
and real on
Show that
  ,
and
for each
Exercise
2.
Let
be a non-constant holomorphic function on a domain
and let
be a domain such that its closure
is compact and contained in
.
Show that, if
 is
constant on the boundary of
 then
there exists at least one zero of
in
(consider
and use a contradiction argument).
Exercise
3.
Let
be a holomorphic function in the disk
 and
put
for
0
Show that
a)
 is
a continuous and increasing function of
in 0
b) If
is not constant, then
is strictly increasing.
Exercise 4.
Let
be holomorphic in
and
put
for
Show that,
if
denotes the
 th
Taylor coefficient of
at
then
deduce
that, if
then
i)
 is
an increasing continuous function of r;
ii)
iii)
 is
a convex function of log
in the case where
is not identically
(show that ,if we put
we have
 to
show that
use the Cauchy-Schwarz inequality on the absolutely
convergent series:
Exercise 5. Let
be a holomorphic function on the disk
such that
on this disk; if there exist two distinct points
and
in the disk such that
then we must have
on the disk.
(consider
the function
 with
from which we have
 and
in the disk).
Exercise 6
Hadamard’s
theorem on the real part: Let
be a holomorphic function in an open set containing the
disk
For
put
i) Show
that
is an increasing continuous function of
(notice that ).
ii) If we
have in addition
 then,
for
we have
(consider
the function
iii) For
show that
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