Can someone prove to me if this inequality with norms and dot products holds?

Question

Can someone show me if the following inequality holds ($\cdot$ is multiplication)?

$\|a\|\cdot\|b\| \le \frac{1}{2}(\|a\|^2 + \|b\|^2)$

I am sure the following holds ($\bullet$ is dot product):

$a\bullet b \le \frac{1}{2}(||a||^2 + ||b||^2)$

Answer 1

Consider $(||a|| - ||b||)^2 = ||a||^2 - 2||a||||b|| + ||b||^2 \geq 0 \Rightarrow ||a||^2 + ||b||^2 \geq 2||a||||b||$