Let X have Poisson(λ) distribution and let Y have Poisson(2λ) distribution.
(i) Prove P (X ≥ Y ) ≤ exp(−(3 − √8)λ) if X and Y are independent.
(ii) Find constants A < ∞, c > 0, not depending on λ, such that, without
assuming independence, P (X ≥ Y ) ≤ A exp(−cλ).

A hint says:

Note that
P (X ≥ Y ) = P (tX ≥ tY ) = P (exp tX ≥ exp tY )
∀ t greater than 0. Now try to bound the right hand side using appropriate
expectation inequalities and optimize over t.

So far I'm trying to do i). I understand what the hint is trying to get at, but I'm not sure which inequalities to use. I tried Jensen's inequality and Chebyshev's inequality but I couldn't get it to work.