I was working on a problem where I had reached a step $$c^2\|x+y\|+ 2 \varepsilon \|T\|^2 \|x\|\|y\|$$ This made me curious and I just wanted to know if there were any inequalities in general that involved $\|x+y\|$ and $\|x\|\|y\|$ that I didn't know about?
The only one I've been able to come up with so far is \begin{align*} (\|x+y\|)^2 &\leq (\|x\|+\|y\|)^2\\ &= \|x\|^2+\|y\|^2+2\|x\|\|y\| \end{align*}
But I wanted to know if there's something more interesting.
Well, in a norm induced by an inner product like the Euclidean norm, you have
$$\| x + y \|^2 = \| x \|^2 + \| y \|^2 + 2 \cos(\theta) \| x \| \| y \|$$
so the only universal inequality you can apply here is the Cauchy-Schwarz inequality which is just $-1 \leq \cos(\theta) \leq 1$ in this notation.