I'm a little stuck with the following inequality and was wondering if anyone could help me out.
at which point I get stuck.
I was also thinking of using the property that:
but the numerators are throwing me off.
For the numerator, use Moo's idea, and for the denominator, use indeed yours (the lower bound in the triangular inequality).
About this lower bound, it is a consequence of the usual triangle inequality: for any , , hence and by symmetry the same holds swapping and , so that .