The Cherno -Borovkov-Utev inequality resulted owing to earlier inequalities established by Cherno (1981) and Borovkov and Utev (1983), respectively, giving bounds for the variance of functions of normal r.v.'s and leading to characterizations of normality. Subsequently, several analytic properties of variance bounds and other relevant results were established by others. DeΓΏning the mean absolute deviation (about a median) as E|X -med(X )| where med(X ) is a median of the distribution of the random variable X , Freimer and Mudholkar (1991) gave a bound for the mean absolute deviation of a certain real-valued function of an absolutely continuous random variable (w.r.t. Lebesgue measure) and Korwar (1991) presented an analogue of this in the discrete case; these authors, also, characterized the Laplace and a mixture of two Waring distributions via the respective bounds.
We extend these latter results theorems to the case where the distributions are not necessarily purely discrete or absolutely continuous, via the approach of Alharbi and Shanbhag (1996). The results in Freimer and Mudholkar (1991) and Korwar (1991) are now corollaries to our ΓΏndings. Also, following Alharbi and Shanbhag (1996), we relate these results to the modiΓΏed version of Cox's representation for a survival function in terms of the hazard measure, given in Kotz and Shanbhag (1980). (The original version of the representation mentioned had appeared in Cox (1972).