I'm not sure about "nice" (after all, one can invent infinitely many inequalities for 3 positive numbers) Yes, of course one can come up with as many random inequalities as one wants. I should have said "Within the context of a larger problem I'm working on, proving this will give us a nice result." Thanks for the solution by the way, very clever.

@ChristianRemling: That maximisation is basically exactly what we did to find the $p \geq \frac{3}{2}$. We would like to prove it, as it would be quite a nice result, but unfortunately it's proven quite difficult.

The inequality implying that p is at least 3/2 is of slightly more interest. The inequality came about in a paper (not by me) about the L^p-improving properties of the convolution operator generated by the Cantor Measure. The inequality holds for some value of p. We got the specific value of p using numerical methods. We are very certain we have the correct values for p, but we are having trouble proving it.