\[\Pr[X\ge m]\le \frac{G_X(a)}{a^m}\] for any \(a>1\) where \(G_X\) is the generating function for the probability distribution of \(X.\)

Similarly, \[\Pr[X\ge m] \le \Pr[a^X \ge a^m] \le \mathop{\mathrm{\mathbb{E}}}[a^X]/a^m.\]

These are basically how you can get Chernoff bounds I guess.

something else

For a non-negative ranvom variable \(X\) \(\mathop{\mathrm{\mathbb{E}}}[X | Y] \le \mathop{\mathrm{\mathbb{E}}}[X] / \Pr[Y]\)