Mergelyan's theorem can be stated as follows (Krantz 1999). Let 
be compact and suppose 
has only finitely many connected components. If 
is holomorphic on the interior of 
and if 
, then there is a rational
function 
with poles in 
such that
 |
(1)
|
A consequence is that if 
is an infinite set of disjoint open
disks 
of radius 
such that the union is almost the unit disk. Then
 |
(2)
|
Define
 |
(3)
|
Then there is a number 
such that 
diverges for 
and converges for 
. The above theorem gives
 |
(4)
|
There exists a constant which improves the inequality, and the best value known is
 |
(5)
|
