Reliability and expectation bounds based on Hardy's inequality

In this article, we provide a probabilistic proof to the strengthened Hardy's integral inequality given in text. We also provide the upper and lower bounds for expectation of functions of hazard rate, mean residual life, eta function and intensity function. Moreover, an upper bound for extropy and cumulative residual extropy is obtained based on Hardy's inequality. Furthermore, we obtain upper bounds for cumulative residual Tsallis entropy of series and parallel systems.