... O(1)mi=0sikβi.(21)2 Journal of Inequalities and Applicationswhere Δ is the forward differ ence operator and tndenotes the nth term of the matrixtransform of the sequence{sn},wheresn:=nk=0ak.Thustn=nk=0anksk=nk=0ankkν=0aν=nν=0aνnk=νank=nν=0anνaν,tn− ... follows from (4)thatβn= O(1/n), and hence thatΔλn= O(1/n) by condition (iv).Proof. Let Tndenote the nth term of the A-transform of the series(anλn)/(nann). Thenwe may writeTn=nν=0anννi=0aiλiaiii=mi=0aiλiaiiinν=ianν=ni=0aniaiλiaiii. ... pn|k, k ≥ 1.Proof. Conditions (iii)–(vii) of Corollary 3 are, respectively, conditions (vi)–(x) of Theo-rem 1 .Conditions (i), (ii), and (v) of Theorem 1 are automatically satisfied for any weightedmean...