Abstract
Necessary and sufficient conditions are given for the validity of discrete Hardy's inequality for the sum of two discrete Hardy-type operators with not necessary non-negative coefficients.
Highlights
Discrete Hardy inequality has been characterized for the discrete Hardy operator in [2,3,9]
Sufficient condition is available in [1] for the validity of the discrete Hardy inequality, for the discrete Hardy-type operator with kernel k = {km,n}; m, n ∈ Z+ defined on D = {(m, n) ∈ Z+ × Z+ : n ≤ m}, km,n is non-increasing in m and non-decreasing in n
|am|pvm, m=1 n=m m=1 holds for all non-negative sequence {an} ∈ lp(vn) and a suitable positive constant C if and only if max(D, E) < ∞ where m q∞
Summary
Discrete Hardy inequality has been characterized for the discrete Hardy operator in [2,3,9]. Sufficient condition is available in [1] for the validity of the discrete Hardy inequality, for the discrete Hardy-type operator with kernel k = {km,n}; m, n ∈ Z+ defined on D = {(m, n) ∈ Z+ × Z+ : n ≤ m}, km,n is non-increasing in m and non-decreasing in n. For the discrete Hardy-type operator with general kernel, sufficient conditions for the validity of discrete Hardy inequality is available in [8] which is both necessary and sufficient when the kernel is of product type. {φi,n}, {ψi,n}; i = 1, 2 and {an} are sequences of real numbers not necessarily non-negative. For complete description of Hardy inequality, we use to refer [5] and monographs [6,7]
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