Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Note on |A|k Summability Factors for Infinite Series | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 86095 8 pages doi 2007 86095 Research Article A Note on A k Summability Factors for Infinite Series Ekrem Savas and B. E. Rhoades Received 9 November 2006 Accepted 29 March 2007 Recommended by Martin J. Bohner We obtain sufficient conditions on a nonnegative lower triangular matrix A and a sequence An for the series y anXn nann to be absolutely summable of order k 1 by A. Copyright 2007 E. Savas and B. E. Rhoades. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. A weighted mean matrix denoted by N pn is a lower triangular matrix with entries pk Pn where pk is a nonnegative sequence with p0 0 and Pn yn 0 pk. Mishra and Srivastava 1 obtained sufficient conditions on a sequence pk and a sequence An for the series y anPnẢn npn to be absolutely summable by the weighted mean matrix N pn .Bor 2 extended this result to absolute summability of order k 1. Unfortunately an incorrect definition of absolute summability was used. In this note we establish the corresponding result for a nonnegative triangle using the correct definition of absolute summability of order k 1 see 3 . As a corollary we obtain the corrected version of Bor s result. Let A be an infinite lower triangular matrix. We may associate with A two lower triangular matrices A and A whose entries are defined by n ank ani ank ank - an-1 k 1 i k respectively. The motivation for these definitions will become clear as we proceed. Let A be an infinite matrix. The series y ak is said to be absolutely summable by A of order k 1 written as A k if X nk-11 Atn-1 k 00 2 k 0 2 Journal of Inequalities and Applications where A is the forward difference operator and tn denotes the nth term of the matrix transform of the sequence sn where sn y n 0 ak. Thus n n k n n n tn
