I have these pair of numbers $ (a, b) = (\frac{4}{9}, \frac{1}{9}) $ and $(c, d) = (\frac{1}{2}, \frac{1}{6}) $. (Number mean nothing, just for illustration and simplification)
Note that - (a, b) are pair of numbers which represent $((E(e_1))^2, (E(e_2))^2) $ and (c, d) are pair for $((E(e_1^2)), (E(e_2^2))) $
Where, E is the expectation.
Clearly, c>a and d>b (by Jensen inequality) But when I divide $\frac{a}{b}$ and $\frac{c}{d}$, I get $\frac{a}{b}$ > $\frac{c}{d}$.
So, does Jensen's inequality flips when two Jensen inequality are divided?
Is there a property which ensure that this will be true always.
Please help