SOME INEQUALITIES RELATED TO THE PRINGSHEIM, STATISTICAL AND σ-CORES OF DOUBLE SEQUENCES

Year 2013, Volume: 42 Issue: 2 , 115 - 123 , 01.02.2013

Celal Çakan Bilal Altay Hüsamettin Coşkun

https://izlik.org/JA47HN36LY

Abstract

SOME INEQUALITIES RELATED TO THE PRINGSHEIM, STATISTICAL AND σ-CORES OF DOUBLE SEQUENCES

https://izlik.org/JA47HN36LY

Abstract

The statistical convergence of double sequences was presented byMursaleen-Edely and Tripathy in two ways, [10, 16]. The statisticalcore of double sequences was introduced by C¸ akan- Altay, [1]. Theσ-convergence and σ-core of double sequences were defined by C¸ akanAltay-Mursaleen, [5]. In this paper, we will study some new inequalities related to the Pringsheim, statistical and σ-cores of double sequences.To achieve this goal, we will characterize some classes offour-dimensional matrices.

Keywords

Double sequences , four dimensional matrices , core theorems and matrixtransformations.

References

• C ¸ akan, C., Altay, B. Statistically boundedness and statistical core of double sequences, J. Math. Anal. Appl. 317 (2), 690–697, 2006.
• C ¸ akan, C., Altay, B. A class of conservative four dimensional matrices, J. Inequalities & Applications Article ID 14721, 1–8, 2006.
• C ¸ akan, C., Altay, B., C ¸ o¸ skun, H. σ-regular matrices and a σ-core theorem for double sequences, Hacettepe J. Math.& Statistics 38 (1), 51–58, 2009.
• C ¸ akan, C., Altay, B., C ¸ o¸ skun, H. On the σ–core of double sequences, preprint. C ¸ akan, C. Altay, B., Mursaleen, The σ-convergence and σ-core of double sequences, Applied Mathematics Letters 19 (10), 1122–1128, 2006.
• Lorentz, G. G. A contribution to the theory of divergent sequences, Acta Math. 80, 167–190, 19 Sava¸ s, E., Patterson, R. F. Lacunary statistical convergence of multiple sequences, Appl. Math. Letters 19, 527–534, 2006.
• Hamilton, H. J. Transformations of multiple sequences, Duke Math. J. 2, 29–60, 1936. Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford Ser. 2 34, 77–86, 1983.
• Mursaleen and Edely, O. H. H. Statistical convergence of double sequences, J. Math. Anal. Appl. 288, 223–231, 2003.
• Patterson, R. F. Double sequence core theorems, Internat. J. Math.& Math. Sci. 22, 785–793, 199 Patterson, R. F., Sava¸ s, E.Uniformly summable double sequences, Studia Sci. Math. Hungar. 44 (1), 147–158, 2007.
• Pringsheim, A. Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53, 289–321, 1900.
• Raimi, R. Invariant means and invariant matrix methods of summability, Duke Math. J. 30, 81–94, 1963.
• Robinson, G. M. Divergent double sequences and series, Trans. Amer. Math. Soc. 28, 50–73, 19 Tripathy, B.C. Statistically convergent double sequences,
• Tamkang J. Math. 34 (3), 231–237, 200 Tripathy, B.C., Sarma, B. On some classes of difference double sequence spaces, Fasciculi Mathematici 41, 135–142, 2009.
• Tripathy, B.C., Sarma, B. Vector valued double sequence spaces defined by Orlicz function, Mathematica Slovaca 59 (6), 767–776, 2009.