Markov inequality vs. Chernoff bound

I have realised that the tightest Chernoff bound for a exponential random variable, X∼exp(λ)X \sim exp(\lambda), is worse than the Markov inequality for xx close 1/λ 1/\lambda . Are there known results as to under what conditions Chernoff bound is tighter than Markov’ s or vice versa?

I have searched a bit for a similar question here or on the web but couldn’ t find an answer.

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