I don't even begin to see how the t-inequalities can be used to prove the m-n-inequalities. But I can see how to prove the m-n-inequalities by integration, using the fact that .

In fact, is an upper Riemann sum for that integral, so that When you multiply through by , that becomes

The other inequality comes in a similar way from the lower Riemann sum obtained by dividing the unit interval into m+1 equal subintervals:

But those t-inequalities don't seem helpful at all.