Some Tauberian theorems for iterations of Holder integrability method

Let f be a real or complex-valued function on [1,infinity) by sigma k(s(x))={1/xt integral 1x sigma k-1(s(t))dt,k >= 1s(x),k=0 for each nonnegative integer k. An improper integral integral 1 infinity f(x)dx is said to be integrable to a finite number mu by the k-th iteration of Holder or Cesaro mean method of order one, or for short, the (H, k) integrable to mu if limx ->infinity sigma k(s(x))=mu. In this case, we write s(x)->mu(H,k). It is clear that the (H, k) integrability method reduces to the ordinary convergence for k=0 and the (H, 1) integrability method is (C, 1) integrability method. It is known that limx ->infinity s(x)=mu implies limx ->infinity sigma k(s(x))=mu. But the converse of this implication is not true in general. In this paper, we obtain some Tauberian conditions for the iterations of Holder integrability method under which the converse implication holds.