Some tauberian conditions for the weighted mean method of summability

Let p = (pn) be a sequence of nonnegative numbers and Pn:= ?n k=0 pk › ? as n › ?. Let the weighted general control modulo of the oscillatory behavior of integer order m ? 1 of a sequence (un) be denoted by (?n,p (m) (u)). We prove that if the weighted generator sequence of a sequence u = (un) of real numbers is summable to a finite number by the weighted mean method, (?(1) n,p(?(m-1) (u))) is increasing, the conditions (Formula presented) are satisfied, and certain conditions on (pn) are hold, then (un) is slowly oscillating. © 2018, Universitatii Al.I.Cuza din Iasi. All rights reserved.