Estimates for harmonic functions

Question

1. Assume $u$ is harmonic in $U$. Then $$ |D^{\alpha}u(x_{0})|\leq \frac{C_{k}}{r^{n+k}}\|u\|_{L^{1}(B(x_{0},r))} $$ for each ball $B(x_{0},r)\subset U$ and each multi-index $\alpha$ of order $|\alpha| = k$.

Here $$ C_{0} = \frac{1}{\alpha_{n}}, C_{k} = \frac{(2^{n+1}nk)^{k}}{\alpha(n)} $$

This is the local estimate for harmonic function. Furthermore, any harmonic function is analytic in the same domain. So we have:

1. Cauchy inequality: $$ |f^{(n)}(z_{0})|\leq n!\frac{\max|f(z)|}{\rho^{n}} $$

I want to know the difference and relationship between this two theorem. (Both of those inequalities can get the Liouville's theorem.)

Answer 1

In most ways, the first one is stronger and more general:

1. It works in all dimensions, not only $n=2$.
2. It uses the mean value of $|u|$ (that is, $r^{-n} \|u\|_{L^{1}(B(x_{0},r))}$) instead of its maximal value.
3. It works for harmonic functions, which are more general than holomorphic functions. (It is straightforward to generalize the concept of harmonicity to complex-valued functions, and the inequality still holds).

However, the second one has the best possible constant $n!$ (equality is attained for $f(z)=z^n$), and I don't think that the first inequality has the sharp constant when $k>0$.

So: you can get the second from the first, but not with the best possible constant.