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Let $X\sim N(0,1)$. I know about Chebyshev's to find an upper bound on $P(X>x)$. I also know about this concentration inequality which is sharper than Chebyshev's for a normal distribution: $P(X>x)\leq \phi(x)/x$ where $\phi(\cdot)$ is the density of standard normal distribution and $x>0$.

But how can I find an upper bound, preferably tight, on $P(0<X<x)$ where $x>0$?

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There is a corresponding lower bound for your sharper inequality: $$\frac{x}{1+x^2} \phi(x) \le P(X > x) \le \frac{1}{x} \phi(x).$$ [See Wikipedia] This can then be converted into an upper bound for $P(0 < X < x)$.

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