It is easy to prove that this implication is false, by considering a simple counter-example courtesy of VincentYu.
Even simpler counter example: X and Y take values 1 and 100, independently and with equal probability.
Then E(X/Y) = 1⁄4 ( 1⁄1 + 1⁄100 + 100⁄100 + 100⁄1), which is basically 25.5. Ditto for E(Y/X).
Expectation is a mean, and in means, large terms dominate—for E(X/Y), the (X=100,Y=1) situation dominates, while for E(Y/X), the (X=1,Y=100) situation dominates.
In fact, if X and Y are independent, identically distributed, strictly positive and non-trivial (ie not just constants), then I think that we always have E(X/Y) > 1.
Even simpler counter example: X and Y take values 1 and 100, independently and with equal probability. Then E(X/Y) = 1⁄4 ( 1⁄1 + 1⁄100 + 100⁄100 + 100⁄1), which is basically 25.5. Ditto for E(Y/X).
Expectation is a mean, and in means, large terms dominate—for E(X/Y), the (X=100,Y=1) situation dominates, while for E(Y/X), the (X=1,Y=100) situation dominates.
In fact, if X and Y are independent, identically distributed, strictly positive and non-trivial (ie not just constants), then I think that we always have E(X/Y) > 1.