Mean of two exponentials versus exponential of mean

For real numbers $a$ and $b$, prove that $\tfrac12\left(e^a+e^b\right)\ge e^{(a+b)/2}$.

The difference is a perfect square. With $u=e^{a/2}$, $v=e^{b/2}$, the left minus the right is $\tfrac12(u^2+v^2)-uv=\tfrac12(u-v)^2\ge0$, with equality iff $a=b$. This is AM-GM, or equivalently convexity of $e^x$ (Jensen).