Generalizations of Some Concentration Inequalities

For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $\phi$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle \phi(\lambda_{1}) \mu(\{x \in \Omega : f(x) \geq \lambda_{1} \}) + \sum_{k=2}^{n}\left(\phi(\lambda_{k})- \phi(\lambda_{k-1})\right) \mu(\{x \in \Omega : f(x) \geq \lambda_{k}\}) ,$ where $0 < \lambda_1 < \lambda_2 \cdots \lambda_n < \infty$. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaym\'e-Chebyshev, Cantelli and Hoeffding inequalities are obtained.