Lower bound for product is a constant?

Question

Assume $c>0$.

I showed that $$ \langle x,x\rangle\geqslant D \exp\Big(-\frac{c}{2}z\Big) $$ and $$ \langle y,y\rangle\geqslant D\exp\Big(\frac{c}{2}z\Big). $$ for some constant $D>0$.

Does this mean that $$ \langle x,x\rangle \langle y,y\rangle\geqslant D^2 \exp(0)=D^2, $$

i.e. the product has a constant positive lower bound?

Answer 1

Yes, $D^2$ is a constant positive lower bound.

Answer 2

Assuming $\langle x,x\rangle \ge 0$, we have $$ \langle x,x\rangle \langle y,y\rangle\geqslant \langle x,x\rangle D\exp(\frac{c} {2}z) \ge D \exp(-\frac{c}{2}z)D\exp(\frac{c} {2}z) = D^2. $$