Does the inequality $ \sqrt{a+b}\geq \sqrt{a/2}+\sqrt{b/2}$ have a name? This is somewhat embarrassing but if $a,b$ are nonnegative real numbers the following seems to hold
$$
\sqrt{a+b}\geq \sqrt{\frac{a}{2}}+\sqrt{\frac{b}{2}}.
$$
Does this inequality have a name?

Squaring both sides we obtain
$$
\sqrt{a+b}\geq \sqrt{a/2}+\sqrt{b/2} \iff a+b\ge \frac{a+b+2\sqrt{ab}}2\iff \frac {a+b}2\ge \sqrt{ab}
$$
which is true by AM-GM and squaring again we obtain
$$\iff \frac{a^2+2ab+b^2}{4}\ge ab \iff a^2-2ab+b^2=(a-b)^2 \ge 0$$
which is a more foundamental result.
