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 2 In 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 ), .
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
Remark 1 Earlier, 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,
This is equivalent to
For this is true, and differentiating gives
which is easily seen to be non-negative. Substitute in (2) will prove the case for general .