what is the best (one with the tightest bounds) concentration inequality for a continuous random variable whose mean and max value is given?

Question

what is the best (one with the tightest bounds) concentration inequality for a continuous random variable whose mean and max value is given? The random variable can take negative values.

Answer 1

By scaling and translation, one can assume that the mean is zero and the max is $1$ (unless the distribution is degenerate). Having done so, splitting the variable into positive and negative parts allows you to apply Markov's inequality to see $P(X \leq -a) \leq \frac{P(X>0)}{a}$ for $a>0$ (since the negative part of $X$ has expectation at most $P(X>0)$.) This then translates into a bound on $P(|X| \geq a)$ once $a>1$. You can then undo the scaling and translation to return to the general case.

Is that the best you can do? Yes: one can look at a variable which is $-a$ with probability $1/a$ and $1$ with probability $1-1/a$ to see that. The continuous aspect does not diminish this, since one can mollify this example arbitrarily little to obtain a continuous example with arbitrarily similar properties.