The (C, ?) integrability of functions by weighted mean methods

Let p(x) be a nondecreasing continuous function on [0,?) such that p(0) = 0 and p(t) › ? as t › ?. For a continuous function f(x) on [0,?), we define We say that a continuous function f(x) on [0,?) is (C, ?) integrable to a by the weighted mean method determined by the function p(x) for some ? > -1 if the limit limt›???(t) = a exists. We prove that if the limit limt›???(t) = a exists for some ? > -1, then the limit limt›? ??+h(t) = a exists for all h > 0. Next, we prove that if the limit limt›???(t) = a exists for some ? > 0 and then the limit limt›???-1(t) = a exists.