Suppose is a non-negative function on the unit disk. Suppose is a holomorphic function such that on . Can we draw an upper bound for ?

After discussing with Yin-Tat about this problem, we’ve come up with the following

Theorem 1 (Yin Tat’s estimate, 2011)With the above assumptions, suppose furthermore , then we have

for all .

Corollary 2In particular, if attains its minimum at with , then

To begin with, let’s recall

Theorem 3 (Schwarz lemma)If is a holomorphic function on the unit disk such that and , then

*Proof:* can be extended to a holomorphic function on . By maximum principle, (first on and then take ), .

*Proof:* (Proof of Yin-Tat’s estimate)

Suppose first that . We want to prove

For technical reason, let us assume also that for all . (This can be done by changing to for first. After proving (1) for , we then let . ) Take any and let (not necessarily minimum) and , so .

Take and . Then and . The composition is a function from the unit disk to itself (i.e. ) such that . We then apply the Schwarz lemma to deduce that

for . Note that , so

Rearranging and applying triangle inequality,

We have shown (1). (Note: the denominator is nonzero. )

In the general case, by replacing with , with and with in the above estimate, we have

So

Remark 1Earlier, I have derived a bound (by using Cauchy integral formula, etc, interested readers may see Yin Tat’s facebook). Let me show that the bound in Yin Tat’s estimate is better than this one. First suppose , and for simplicity is real, we have to show for fixed,For this is true, and differentiating gives

which is easily seen to be non-negative. Substitute in (2) will prove the case for general .

It looks like 12.5 in real and complex rudin

Basically, all estimate involving Schwarz lemma or alike uses normalization like this.

The equation after “Note that {\psi(-a)=0}, so” should be “{|v(-a)|=|\phi\circ u(0)|}…” right?

gosh, i should type $\psi(-a)=0$ and $“{|v(-a)|=|\phi\circ u(0)|$…

gosh^2, I am such a noob. I should type and …

Hope this time I am correct.

=.=…

… [Ok, get it, thanks! -KKK (By the way your icon is very cool, I hope I can have this one too ^.^)]

haha thx, btw can you add me to the list of authors?

[Please send me

your email addressafter your have registered a user name, send it to kkkwong@math… -KKK]sent. [I’ve added you to be an administrator. -KKK]

thx=]