I have a little problem with understanding some probably very easy inequality in the proof of Picard's Theorem in Ransford's book (Potential Theory in the Complex Plane). If anyone could show me why this is correct, I would be grateful:
Let $f:\mathbb{C}\to \mathbb{C}$ such that $\mathbb{C}\setminus f(\mathbb{C})$ contains at least two points, $\alpha, \beta$. Put $h:=\log |f-\alpha|$ and $k:=\log|f-\beta|$ and $h^+=\max (h,0)$, $k^+=\max (k,0)$. Obviously $h$ and $k$ are harmonic on $\mathbb{C}$, and we have to justify that
$|h^+-k^+|\leq |\alpha-\beta|$.
The function $t\mapsto \log^+t$ is Lipschitz with constant $1$ on $[0,\infty)$, because its derivative is bounded by $1$ on $(1,\infty)$, and the function is constant on $[0,1]$.
Consequently, $z\mapsto \log^+|z|$ is $1$-Lipschitz on $\mathbb{C}$, as the composition of two $1$-Lipschitz functions. Written out, this means $$ |\log^+|f-\alpha|-\log^+|f-\beta||\le |\alpha-\beta| $$