Abstract:
For a real-valued measurable function f and a nonnegative, nondecreasing function φ, we
first obtain a Chebyshev type inequality which provides an upper bound for φ(λ1)μ({x ∈
Ω : f (x) ≥ λ1}) +
n
∑
k=2
(φ(λk) − φ(λk−1)) μ({x ∈ Ω : f (x) ≥ λk}), where 0 < λ1 < λ2 <
· · · < λn < ∞. Using this, generalizations of a few concentration inequalities such as
Markov, reverse Markov, Bienaymé–Chebyshev, Cantelli and Hoeffding inequalities are
obtained.
