The paper starts with the intimate connections among four classical inequalities, including Hardy’s inequality \[ \sum_{n=1}^{\infty} \left( \frac 1 n \sum_{k=1}^n x_k \right)^p \leq \left(\frac p{p-1}\right)^p \sum_{k=1}^{\infty} x_k^p, \] Carleman’s inequality, Copson’s inequality, and Knopp’s inequality. From a generalization of Carleman’s inequality, known as Love’s inequality, a related generalization of Hardy’s inequality is presented as
\[ \sum_{n=1}^{\infty} n^{\beta -1} \left( \frac 1{n^{\alpha}} \sum_{k=1}^n (k^{\alpha}- (k-1)^{\alpha}) x_k \right)^p \leq \left( \frac {\alpha p}{\alpha p - \beta}\right)^p \sum_{k=1}^{\infty} k^{\beta -1} x_k^p, \]
where \(\alpha >0\), \(\beta\geq 1\), \(p\geq 1\) and \(\alpha p >\beta\). The constant is the best one. The paper then estimates the increment of the sequence \({n^{\alpha}}/(n^{\alpha}- (n-1)^{\alpha})\) and combines it with an inequality of Cartlidge, involving the weighted mean \((\sum_{k=1}^n a_k x_k)/ (\sum_{k=1}^n a_k)\), to show a partial proof of the generalized Hardy’s inequality in the case \(\alpha \geq 1\) and \(\beta = 1\). To prove the inequality completely the paper mainly uses the help of the sequence \(({1^{\alpha}+2^{\alpha}+ \dots + n^{\alpha}})/{n^{\alpha}}.\) The concavity of the sequence and its applications on series, sums of powers and the power majorization problem are extensively studied. Improvements and generalizations on some known results in the fields are given. The generalizations of the four classical inequalities involving the weighted mean operator are studied in the paper too.
Finally, the paper reviews the classical inequalities and studies generalizations of Copson’s inequality and Knopp’s inequality. Like the generalization of the Hardy’s inequality, these generalizations are analogous to that of Love’s inequality from Carleman’s inequality. A conjecture of a generalized Carleman’s inequality is given at the end of the paper.