Markov’s inequality is very general and hence very weak. Assume that X is a non-negative random variable, a > 0, and X has a finite expected value, Then Markov’s inequality says that
In  the author gives two refinements of Markov’s inequality which he calls Hansel and Gretel.
and Gretel says
 Joel E. Cohen. Markov’s Inequality and Chebyshev’s Inequality for Tail Probabilities: A Sharper Image. The American Statistician, Vol. 69, No. 1 (Feb 2015), pp. 5-7