Let $F(z)=z\left\lvert z\right\rvert^a$ for $a\geqslant0$. Show that it is local Lipischitz on $\mathbb{C}$ and that $$\left\lvert F(z)-F(z')\right\rvert\leqslant C_a\left\lvert z-z'\right\rvert\left(\left\lvert z\right\rvert^a+\left\lvert z'\right\rvert^a\right)$$
My work It is of class $\mathcal{C}^1$ on $\mathbb{C}\setminus\left\{0\right\}$ so it is local Lipschitz, however I cannot prove the inequality.
Let $a>0$ as otherwise, the required inequality is trivial and also assume $zz' \ne 0$ for the same reason; applying the mean value theorem to $x^a$ on we get that if $0<x \le y$ then $|x^a-y^a|=y^a-x^a=ac^{a-1}|x-y| \le a\max \{x^{a-1}, y^{a-1}\}|x-y|$ since $c \in [x,y]$
So $|z|z|^a-z'|z'|^a|\le|z-z'||z|^a + |z'(|z|^a-|z'|^a)| \le |z-z'||z|^a+a||z|-|z'|||z'|\max \{|z|^{a-1}, |z'|^{a-1}\}\le |z-z'|(|z|^a+a|z'|\max \{|z|^{a-1}, |z'|^{a-1}\}$
Repeating same with $z,z'$ switched we get that:
$|z|z|^a-z'|z'|^a|\le |z-z'|(|z'|^a+a|z|\max \{|z|^{a-1}, |z'|^{a-1}\}$
But now clearly either $|z'|\max \{|z|^{a-1}, |z'|^{a-1}\}=|z'|^a$ or $|z|\max \{|z|^{a-1}, |z'|^{a-1}\}=|z|^a$, while $|z'|^a, |z|^a \le |z'|^a +|z|^a$ so in any event we get:
$|z|z|^a-z'|z'|^a|\le |z-z'|(a+1)(|z'|^a +|z|^a)$ so we can take $C_a=a+1$