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On Deferred Statistical and Strong Deferred Cesàro Convergences of Sequences With Respect to A Modulus Function

Year 2023, Volume: 44 Issue: 4, 753 - 757, 28.12.2023

Cemal Belen , Mustafa Yıldırım

https://doi.org/10.17776/csj.1334082

Abstract

Let f be any modulus function. We prove that the classes of strongly deferred Cesàro convergent sequences defined by f and deferred statistical convergent sequences are equivalent if the sequence is f-deferred uniformly integrable. Some converse inclusions are obtained when the modulus function f is compatible. Finally, for any compatible modulus f, we prove that any sequence is f-strongly deferred Cesàro convergent if and ony if it is deferred f-statistically convergent and deferred uniformly integrable.

Keywords

Deferred statistical convergence Strong deferred convergence Uniformly integrable sequence Compatible modulus function.

References

  • [1] Fast H., Sur la convergence statistique. Colloq. Math., 2 (3/4) (1951) 241–244. [2] Buck R. C., Generalized asymptotic density, Amer. J. Math., 75 (1953) 335–346.
  • [3] Schoenberg I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959) 361–375.
  • [4] Šalát T., On statistically convergent sequences of real numbers. Math. Slovaca, 30(2) (1980) 139-150.
  • [5] Fridy J. A., On statistical convergence, Analysis, 5 (1985) 301–313.
  • [6] Maddox I.J., Sequence spaces defined by a modulus, Math. Proc., Cambridge Philos. Soc., 100 (1986) 161-166.
  • [7] Connor J., On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull, 32 (1989) 194-198.
  • [8] Connor J., The statistical and strong p-Cesaro convergence of sequences, Analysis, 8 (1988) 47-63.
  • [9] Khan M. K. and Orhan C., Matrix characterization of A-statistical convergence, J. Math. Anal. Appl. 335 (2007) 406–417.
  • [10] Aizpuru A., Listàn-Garc13 ̆053'fa, M. C., Rambla-Barreno, F., Density by moduli and statistical convergence, Quaest. Math., 37 (4) (2014) 525-530.
  • [11] León-Saavedra F., del Carmen Listán-Garc13 ̆053'fa M., Fernández F. J. P., de la Rosa M. P. R. On statistical convergence and strong Cesàro convergence by moduli. J. Inequal. Appl., (2019) 2019:298.
  • [12] Agnew R. P., On deferred Cesàro means. Annals of Mathematics, (1932) 413-421.
  • [13] Küçükaslan M. and Yılmaztürk M., On deferred statistical convergence of sequences, Kyungpook Math. J, 56(2) (2016) 357-366.
  • [14] Gupta S. and Bhardwaj V. K., On deferred f-statistical convergence, Kyungpook Math., J. 58 (2018) 91-103.
  • [15] Cinar M., Yilmaz E. and Et M. Deferred statistical convergence on time scales. Proceedings of the Romanian Academy, Series A:, 22 (4) (2021) 301–306.
  • [16] Dagadur I. and Sezgek S., Deferred Cesàro mean and deferred statistical convergence of double sequences. J. Inequal. Spec. Funct, 4 (2016) 118-136. [17] Et M. and Yilmazer M. C., On deferred statistical convergence of sequences of sets. AIMS Mathematics, 5 (3) (2020) 2143-2152.
  • [18] Et M., Baliarsingh P., Kandemir H. Ş. and Küçükaslan M., On μ-deferred statistical convergence and strongly deferred summable functions. RACSAM, 115 (1) (2021) 34 1-14.
  • [19] Kişi Ö., Gürdal M. and Savaş E., On Deferred Statistical Convergence of Fuzzy Variables. Applications and Applied Mathematics: An International Journal (AAM), 17 (2) (2022) 5.
  • [20] Kişi Ö. and Gürdal M., On deferred Cesàro summability and statistical convergence for the sets of triple sequences, Annals of Fuzzy Mathematics and Informatics, 24(2) (2022) 115–127.
  • [21] Kişi Ö. and Gürdal M., Certain aspects of deferred statistical convergence of fuzzy variables in credibility space, The Journal of Analysis, (2023) 1-19.
  • [22] Ulusu U., and Gülle E., Deferred Cesàro summability and statistical convergence for double sequences of sets, Journal of Intelligent & Fuzzy Systems, 42(4) (2022) 4095-4103. [23] Nakano H., Concave modulars, J. Math Soc. Japan, 5(1) (1953) 29-49.
  • [24] Belen C., Yıldırım M. and Sümbül C., On statistical and strong convergence with respect to a modulus function and a power series method. Filomat, 34 (12) (2020) 3981-3993.
  • [25] León-Saavedra F., del Carmen Listán-Garc13 ̆053'fa M., Fernández F. J. P., de la Rosa M. P. R. Correction to: On statistical convergence and strong Cesàro convergence by moduli. J. Inequal. Appl., J Inequal Appl 2023, 110 (2023). https://doi.org/10.1186/s13660-023-02988-0.

Year 2023, Volume: 44 Issue: 4, 753 - 757, 28.12.2023

Cemal Belen , Mustafa Yıldırım

https://doi.org/10.17776/csj.1334082

Abstract

References

  • [1] Fast H., Sur la convergence statistique. Colloq. Math., 2 (3/4) (1951) 241–244. [2] Buck R. C., Generalized asymptotic density, Amer. J. Math., 75 (1953) 335–346.
  • [3] Schoenberg I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959) 361–375.
  • [4] Šalát T., On statistically convergent sequences of real numbers. Math. Slovaca, 30(2) (1980) 139-150.
  • [5] Fridy J. A., On statistical convergence, Analysis, 5 (1985) 301–313.
  • [6] Maddox I.J., Sequence spaces defined by a modulus, Math. Proc., Cambridge Philos. Soc., 100 (1986) 161-166.
  • [7] Connor J., On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull, 32 (1989) 194-198.
  • [8] Connor J., The statistical and strong p-Cesaro convergence of sequences, Analysis, 8 (1988) 47-63.
  • [9] Khan M. K. and Orhan C., Matrix characterization of A-statistical convergence, J. Math. Anal. Appl. 335 (2007) 406–417.
  • [10] Aizpuru A., Listàn-Garc13 ̆053'fa, M. C., Rambla-Barreno, F., Density by moduli and statistical convergence, Quaest. Math., 37 (4) (2014) 525-530.
  • [11] León-Saavedra F., del Carmen Listán-Garc13 ̆053'fa M., Fernández F. J. P., de la Rosa M. P. R. On statistical convergence and strong Cesàro convergence by moduli. J. Inequal. Appl., (2019) 2019:298.
  • [12] Agnew R. P., On deferred Cesàro means. Annals of Mathematics, (1932) 413-421.
  • [13] Küçükaslan M. and Yılmaztürk M., On deferred statistical convergence of sequences, Kyungpook Math. J, 56(2) (2016) 357-366.
  • [14] Gupta S. and Bhardwaj V. K., On deferred f-statistical convergence, Kyungpook Math., J. 58 (2018) 91-103.
  • [15] Cinar M., Yilmaz E. and Et M. Deferred statistical convergence on time scales. Proceedings of the Romanian Academy, Series A:, 22 (4) (2021) 301–306.
  • [16] Dagadur I. and Sezgek S., Deferred Cesàro mean and deferred statistical convergence of double sequences. J. Inequal. Spec. Funct, 4 (2016) 118-136. [17] Et M. and Yilmazer M. C., On deferred statistical convergence of sequences of sets. AIMS Mathematics, 5 (3) (2020) 2143-2152.
  • [18] Et M., Baliarsingh P., Kandemir H. Ş. and Küçükaslan M., On μ-deferred statistical convergence and strongly deferred summable functions. RACSAM, 115 (1) (2021) 34 1-14.
  • [19] Kişi Ö., Gürdal M. and Savaş E., On Deferred Statistical Convergence of Fuzzy Variables. Applications and Applied Mathematics: An International Journal (AAM), 17 (2) (2022) 5.
  • [20] Kişi Ö. and Gürdal M., On deferred Cesàro summability and statistical convergence for the sets of triple sequences, Annals of Fuzzy Mathematics and Informatics, 24(2) (2022) 115–127.
  • [21] Kişi Ö. and Gürdal M., Certain aspects of deferred statistical convergence of fuzzy variables in credibility space, The Journal of Analysis, (2023) 1-19.
  • [22] Ulusu U., and Gülle E., Deferred Cesàro summability and statistical convergence for double sequences of sets, Journal of Intelligent & Fuzzy Systems, 42(4) (2022) 4095-4103. [23] Nakano H., Concave modulars, J. Math Soc. Japan, 5(1) (1953) 29-49.
  • [24] Belen C., Yıldırım M. and Sümbül C., On statistical and strong convergence with respect to a modulus function and a power series method. Filomat, 34 (12) (2020) 3981-3993.
  • [25] León-Saavedra F., del Carmen Listán-Garc13 ̆053'fa M., Fernández F. J. P., de la Rosa M. P. R. Correction to: On statistical convergence and strong Cesàro convergence by moduli. J. Inequal. Appl., J Inequal Appl 2023, 110 (2023). https://doi.org/10.1186/s13660-023-02988-0.
There are 22 citations in total.

Details

Primary Language English
Subjects Theoretical and Applied Mechanics in Mathematics
Journal Section Natural Sciences
Authors

Cemal Belen 0000-0002-8832-1524

Mustafa Yıldırım 0000-0002-8880-5457

Publication Date December 28, 2023
Submission Date July 28, 2023
Acceptance Date November 2, 2023
Published in Issue Year 2023Volume: 44 Issue: 4

Cite

APA Belen, C., & Yıldırım, M. (2023). On Deferred Statistical and Strong Deferred Cesàro Convergences of Sequences With Respect to A Modulus Function. Cumhuriyet Science Journal, 44(4), 753-757. https://doi.org/10.17776/csj.1334082
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