Does Jensen's Inequality hold for complex numbers?

Question

We can use Jensen's inequality to show that if $1<p<\infty$, then there exists a constant $C>0$ such that for every $x,y \in \mathbb{R}$, we have: $$ |x+y|^p \leq C (|x|^p + |y|^p) $$ Can we show this inequality for complex numbers? If so, can we use Jensen's inequality to show it? According to wikipedia, the usual definition of convexity is only for real vector spaces.

Answer 1

For complex numbers $z, w$ $$ |z+w|^p \le (|z| + |w|)^p \, . $$ Now apply your inequality to the real numbers $x = |z|$ and $y = |w|$: $$ (|z| + |w|)^p \le C (|z|^p + |w|^p) \, . $$