1.   Prove that

is attained at a point of is constant on  

is equivalent to

has a relative maximum at ais constant in a neighborhood of a  

2. Let f(z) be a non-constant analytic function on the disk  with . Show that on every circle assumes both positive and negative values.

3.       Suppose that   f(z) is an analytic function on  a domain D which contains a simple closed curve   and the inside of .  If |f| is constant on , then f is either constant or has a zero inside .

4.  Suppose that f(z) is an analytic function whose modulus |f(z)| is constant on  a domain D.  Show that f(z) is constant.

(Hints: With  is constant.  Differentiating with respect to x and y we get .  Using the Cauchy Riemann equations we get

.  If , then holds only at individual points, so that must hold everywhere and therefore .)

5.   Let  be a bounded domain, and consider  points  in   Show that the product

 of distances from a point  varient in the closure,
 to the points  attains its maximum at a point on the boundary of

6.  Three-Circle Problem of Hadamard:  Let  be a holomorphic function in an open set containing the closed annulus   Put  for  

Show that

(1)              

for Apply the maximum principle on the function  with  integers and  choose then a real number  such that  and a sequence of integers ()  such that   as .

Verify that inequality (1) implies that is a convex function of  for