The connection with was as follow. Let be the graph of your favorite function. First define the infimum of “visually” by saying that “we draw a horizontal line under the graph and then raise it as far as we can until it bumps into the graph of . The can be defined analogously by drawing a horizontal line under the graph and continue raising it as long as the set of ‘s where is under the line forms a bounded set.

Proof: Unfold and rewrite what we want to show as:

By Cauchy-Schwarz the left hand-side is less than

which can be written as

But, since ,

determine all possible triples of non-negative integers such that

There are some trivial solutions. Whenever for some other non-negative integer , then is a solution.

Are there any other? (I haven’t put much thought to this question).

An equivalent definition of the spaces in the half-plane, and some applications.

and

Then

for some constants .

.

One might wonder whether such a global property occurs for other shapes, say ellipses. The answer is no.

Fact: Suppose is open and has finite area , , and

for every integrable harmonic function on . Then is a disk centered at .

To see why, pick a point that is closest to . Then play with the function

Let be entire and suppose that for all :

Show that .

Solution: For let

On one hand, using the fact that

we get that as , for each .

And on the other hand, direct integration shows that where . .

Questions: Is there a different proof, say, using the maximum principle? What if is simply harmonic?

Update: The same power series approach seems to give the harmonic case.