Prove of complex numbers inequality $|a\bar{b}| \leq |a|^2 + |b|^2$

Question

Is it true that for any two complex numbers, say $a, b$, the following inequality holds: $|a\bar{b}| \leq |a|^2 + |b|^2$ ? How can we prove this?

Answer 1

We have

$$|\overline b|=|b|$$ and from $$(|a|-|b|)^2\ge0$$ we get $$|ab|\le \frac12(|a|^2+|b|^2)$$