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

Abstract

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

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

1. C ¸ akan, C., Altay, B. Statistically boundedness and statistical core of double sequences, J. Math. Anal. Appl. 317 (2), 690–697, 2006.
2. C ¸ akan, C., Altay, B. A class of conservative four dimensional matrices, J. Inequalities & Applications Article ID 14721, 1–8, 2006.
3. 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.
4. 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.
5. 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.
6. 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.
7. Mursaleen and Edely, O. H. H. Statistical convergence of double sequences, J. Math. Anal. Appl. 288, 223–231, 2003.
8. 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.

9. Pringsheim, A. Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53, 289–321, 1900.
10. Raimi, R. Invariant means and invariant matrix methods of summability, Duke Math. J. 30, 81–94, 1963.
11. Robinson, G. M. Divergent double sequences and series, Trans. Amer. Math. Soc. 28, 50–73, 19 Tripathy, B.C. Statistically convergent double sequences,
12. 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.
13. Tripathy, B.C., Sarma, B. Vector valued double sequence spaces defined by Orlicz function, Mathematica Slovaca 59 (6), 767–776, 2009.