moments of eigenvalues?

Question

Let m1, m2, m3 be the moments of [l1, l2, l3] of the order of 1,2,3, where l1>=l2>=l3>=0.
Let y = sqrt(3m2/(2(m1^2+m2)))
and z = sqrt(2)*m3(m2^(-1.5))
How to find an expression for the upper bound of y in terms of z?
Some results I current have are max(y)=1/sqrt(2) when z=-1, max(y) = 1 when z = 1. I have used Monte Carlo approach to plot the shape of max(y) vs z, and it is a strictly increasing upward-concave curve.

Answer 1

Close, it should be an increasing upward convex curve. Try using the value of z=sqrt(-1)