Let $X > 0$ be a random variable; let $P$ be the underlying probability measure; let $\delta > 0$. I wonder if there is already in probability literature a known result giving a sharper bound for $P ( X > \delta )$ than that given by the Chebyshev's. (By Chebyshev's inequality I simply mean the relation $P(X > \delta) \leq \delta^{-1}EX$; one may call it Markov's inequality, which does not affect the discussion here.)
A major motive for this question is that, besides application convenience, there is a result in probability theory (in Chung's probability text, for example) stating that $\sum_{n \in \mathbb{N}}P(X > n) \leq EX \leq 1 + \sum_{n \in \mathbb{N}}P(X > n)$. This is stunning, as if $X$ is integrable-$P$ then an upper bound for $P(X > n)$ in terms of $E X$ becomes "infinitely much" sharper than that given by Chebyshev's.
This triggers my interest as to if there is already a result that may be already well-known for probability theorists but somehow less known for applied probability people. If possible, I would love to be guided to the original and/or related literature.
