Abstract:

We give reciprocal versions of the Sclove et al. and Feller inequalities for moments of nonnegative random variables. Our results apply to any nonnegative random variable. The strongest assumption is that the moments involved must be finite. Thus, the results obtained also hold for any empirical distribution with nonnegative data. These facts allow potential applications in numerical analysis, probability, and statistical inference, among other disciplines. Moreover, the proposed methodology offers an alternative approach to obtain new inequalities and even to improve some known inequalities. For instance, we give new inequalities for the ratio of gamma functions. In this context, we also improve an inequality by Bustoz and Ismail and some cases of inequalities due to Gurland and Dragomir et al. Additionally, we present a new inequality for finite sums of nonnegative or nonpositive numbers. For some cases, this relation improves even the Cauchy–Bunyakovsky–Schwarz inequality.

2010 AMS Mathematics Subject Classification: Primary 60E15; Secondary 26D15, 33B15, 11L07.

Keywords and phrases: moment inequalities, ratio of gamma functions, alternative expectation formula, alternative covariance formula.