... (3.16)holds for some s∈ (0,1/p), then the inequality (1.3)holdswithC≤ inf0<s<1/pp − 1p − sp− 11/pE(s). (3.17)Remark 3.8. For the case dn,k= 1, n,k = 1,2, , the condition ... ith the additional condition that {vn}∞n=1isnonincreasing if and only if the condition (2.4) holds. Moreover, for the best constant C in(3.18), the estimate (2.8)holds.Remark 3.10. For the ... the corresponding Hardy- type inequalities for general nonnegative sequences. Hence, theycan be characterized by the same condition(s); see the books [6, 9]butalsothemorerecent results, for example,...