Open Access

Subsequential Convergence Conditions

Journal of Inequalities and Applications20072007:087414

DOI: 10.1155/2007/87414

Received: 27 April 2007

Accepted: 19 August 2007

Published: 9 October 2007

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq1_HTML.gif be a sequence of real numbers and let https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq2_HTML.gif be any https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq3_HTML.gif regular limitable method. We prove that, under some assumptions, if a sequence https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq4_HTML.gif or its generator sequence https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq5_HTML.gif generated regularly by a sequence in a class https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq6_HTML.gif of sequences is a subsequential convergence condition for https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq7_HTML.gif , then for any integer https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq8_HTML.gif , the https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq9_HTML.gif repeated arithmetic means of https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq11_HTML.gif , generated regularly by a sequence in the class https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq12_HTML.gif , is also a subsequential convergence condition for https://static-content.springer.com/image/art%3A10.1155%2F2007%2F87414/MediaObjects/13660_2007_Article_1755_IEq13_HTML.gif .

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Authors’ Affiliations

(1)
Department of Mathematics, Adnan Menderes University
(2)
Department of Mathematics, Rockford College

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Copyright

© İbrahim Çanak et al. 2007

This article is published under license to BioMed Central Ltd. 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.