Bounding a holomorphic function on a half disk Question 7.5 from this list is the following.

Suppose you have a holomorphic function on the upper half of the unit
  disk and you know that is absolute value is at most 2 on the
  semi-circle, and at most 1 on the real-axis part of the boundary.
  Bound the function in the interior.

I'm at a loss for how do better than the crude bound of $2$ over the entire domain due to the maximum principle. I was thinking of conformally mapping the domain to a sector and trying to apply a Phragmén-Lindelöf-type theorem, but those don't seem to give anything better. Can we obtain a better estimate?
 A: I'm at a loss for how do better than the crude bound of 2 over the entire domain
Luckily it appears you didn't remain at loss for too long... heh. Yes, we can do what you said. There are at least two things you might have meant by what you said in your auto-answer; one's better than the other.
Say $u$ is the harmonic(!) function that equals $2$ on the semicircle and $1$ on the segment. Since $|f|$ is subharmonic you get $$|f(z)|\le u(z).$$
Now let $v=\log(2)(u-1)$. Then $v=\log(2)$ on the semicircle and $v=0=\log(1)$ on the segment, so since $\log|f|$ is subharmonic you get $$|f(z)|\le e^{\log(2)(u(z)-1)}.$$It's an amusing exercise to show that the second estimate is better; you need to show that $$\log(2)(t-1)<\log(t)\quad(1<t<2).$$(Alternately one could argue somehow that the estimate using $\log|f|$ must be better, giving the world's most obscure proof of that last inequality.)

You mention Phragmén-Lindelöf here and the Three Lines Theorem in that closely related thread. Seems worth pointing out that those results all follow from the fact that $\log|f|$ is subharmonic, together with estimates on harmonic measure. In the traditional P-L thing you concoct a holomorphic function $g$ adapted to the domain and then apply Maximum Modulus to $$F=gf.$$That's the same as applying the subharmonic maximum principle to $\log|F|$.
OR you could say equivalently that it's just using the fact that $\log|f|$ is subharmonic, while using $\log|g|$ to obtain information about harmonic measure.
The proof using subharmonic functions is better. Example: Typically one proves the Three Lines Theorem by a P-Lish approach as above, then derives the Three Circles Theorem by applying an exponential mapping. The P-L argumment doesn't apply directly to the Three Circles Theorem because the appropriate $g$ is not single-valued. But $|g|$ is single-valued, $\log|g|$ is subharmonic, and there you are.
So. The traditional P-L approach to Three Lines is equivalent to using $\log|f|$ subharmonic. The latter also works for Three Circles, while the former does not. The latter is better, qed.
A: If we explicitly compute the function that is $2$ on the semicircle and $1$ on the axis, this will provide a pointwise bound inside the domain.
