Some Improvements on Markov's Theorem with Extensions

Markov's theorem for an upper bound of the probability related to a nonnegative random variable has been improved using additional information in almost the nontrivial entire range of the variable. In the improvement, Cantelli's inequality is applied to the square root of the original variable, whose expectation is finite when that of the original variable is finite. The improvement has been extended to lower bounds and monotonic transformations of the original variable. The improvements are used in Chebyshev's inequality and its multivariate version.