Skip to content


  • Research Article
  • Open Access

Subsequential Convergence Conditions

Journal of Inequalities and Applications20072007:087414

  • Received: 27 April 2007
  • Accepted: 19 August 2007
  • Published:


Let be a sequence of real numbers and let be any regular limitable method. We prove that, under some assumptions, if a sequence or its generator sequence generated regularly by a sequence in a class of sequences is a subsequential convergence condition for , then for any integer , the repeated arithmetic means of , , generated regularly by a sequence in the class , is also a subsequential convergence condition for .


  • Generator Sequence
  • Real Number
  • Convergence Condition
  • Limitable Method
  • Arithmetic Means


Authors’ Affiliations

Department of Mathematics, Adnan Menderes University, Aydin, 09010, Turkey
Department of Mathematics, Rockford College, 5050 E. State Street, Rockford, IL 61108, USA


  1. Stanojević ČV: Analysis of divergence: applications to the Tauberian theory, Graduate Research Seminar. University of Missouri-Rolla, Rolla, Mo, USA, 1999.Google Scholar
  2. Stanojević ČV: Analysis of Divergence: Control and Management of Divergent Process, edited by İ. Çanak, Graduate Research Seminar Lecture Notes. University of Missouri-Rolla, Rolla, Mo, USA; 1998.Google Scholar
  3. Hardy GH: Divergent Series. The Clarendon Press, Oxford University Press, New York, NY, USA; 1949:xvi+396.MATHGoogle Scholar
  4. Boos J: Classical and Modern Methods in Summability, Oxford Mathematical Monographs. Oxford University Press, Oxford, UK; 2000:xiv+586.Google Scholar
  5. Dik M: Tauberian theorems for sequences with moderately oscillatory control modulo. Mathematica Moravica 2001, 5: 57–94.MATHGoogle Scholar
  6. Dik F: Tauberian theorems for convergence and subsequential convergence with moderately oscillatory behavior. Mathematica Moravica 2001, 5: 19–56.MATHGoogle Scholar
  7. Tauber A: Ein Satz aus der Theorie der unendlichen Reihen. Monatshefte für Mathematik und Physik 1897,8(1):273–277. 10.1007/BF01696278MathSciNetView ArticleMATHGoogle Scholar
  8. Littlewood JE: The converse of Abel's theorem on power series. Proceedings of the London Mathematical Society 1911,9(2):434–448.View ArticleMATHGoogle Scholar
  9. Rényi A: On a Tauberian theorem of O. Szász. Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum 1946, 11: 119–123.MATHGoogle Scholar
  10. Çanak İ, Totur Ü: Tauberian theorems for Abel limitability method. submitted for publication submitted for publicationGoogle Scholar
  11. Çanak İ: Tauberian theorems for a generalized Abelian summability methods. Mathematica Moravica 1998, 2: 21–66.MATHGoogle Scholar
  12. Çanak İ, Totur Ü: A note on Tauberian theorems for regularly generated sequences. submitted for publication submitted for publicationGoogle Scholar
  13. Çanak İ, Totur Ü: A Tauberian theorem with a generalized one-sided condition. Abstract and Applied Analysis 2007, 2007: 12 pages.View ArticleGoogle Scholar


© 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.