My question is essentially the same as An inequality concerning an harmonic function , however I did not find the answer given satisfactory. To restate it, I would like to solve the following:
Let $h$ be a positive harmonic function on $K(0, \rho) = \{ z \in \mathbb{C} \ \vert \ \vert z \vert < \rho \}$. Show that
$$ \vert \nabla h(0) \vert \leq \frac{2}{\rho} h(0) $$ and deduce that $$ \vert \nabla h(z) \vert \leq \frac{2\rho}{\rho^2- \vert z \vert^2} h(z) \qquad \quad , \vert z \vert < \rho $$
Showing that $ \vert \nabla h(0) \vert \leq \frac{2}{\rho} h(0) $ was rather easy using Harnacks inequality, my issue comes when trying to prove the last bit. Supposedly we should be able to define an automorphism $\varphi: K(0,\rho) \to K(0,\rho)$ such that $\varphi(0)= z$ and hence use the first result on $h \circ \varphi$, since if $\varphi$ is holomorphic $h\circ \varphi$ is again positive harmonic. Now my problem is that I fail to see why doing so wouldn't just imply that
$$ \vert \nabla h(z) \vert \leq \frac{2}{\rho} h(z) $$ which is not quite what we want.