Some generalizations and improvements of discrete Hardy’s inequality

One-dimensional discrete Hardy and Rellich inequalities on integers

“…We would like to mention that inequality (2.3) was proved in paper [22] with the weight pn ´1{2q α and α P p0, 1q. It is also worthwhile to notice that the above inequalities reduce to corresponding Hardy inequalities on N 0 :" t0, 1, 2, ..u when we restrict ourselves to functions u taking value zero on negative integers.…”

“…Till date, various proofs of Hardy inequalities exist, most recent ones being [13], [6], [20], [18], [17]. It is worthwhile to mention papers [8], [11], [14], [2], [7], [4], [23], [22], [3], [14], [15], [16], where various variants of inequality (1.1) have been studied and applied.…”

Discrete weighted Hardy Inequality in 1-D

“…Recently the method used in [9] was exploited to prove some new discrete hardy inequalities on regular trees in [7]. Before getting into main setting of the paper, we would like to quote papers [13], [16], [6], [15], [14], [8], [21] where various variants of (1.2) are considered, improved and applied.…”

Some Refinements and Generalizations of I. Schur Type Inequalities

“…It is well known that x n = (1 + (1/ n )) n and y n = (1 + (1/ n )) n +1 are, respectively, monotone increasing and monotone decreasing, and both of them converge to the constant e . In fact, extensive researches for the estimated value of e have been studied [ 1 – 4 ], and the methods for estimating the value of e are of benefit to the improvements of the Hardy inequality, Carleman inequality, Gamma function inequality, and so forth [ 5 – 13 ], which is an essential motivation for this work. Klambauer and Schur have reached the following conclusion.…”