Korean J. Math. Vol. 29 No. 1 (2021) pp.169-177
DOI: https://doi.org/10.11568/kjm.2021.29.1.169

On deferred Ces\`{a}ro mean in paranormed spaces

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Sinan Ercan


The aim of the present study is to introduce the concepts of deferred statistical convergence, deferred statistical Cauchy sequence and deferred Ces\`{a}ro summability in paranormed spaces. We investigate some properties of these concepts and some inclusion relations with examples.

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[1] R. P. Agnew, On deferred Ces aro Mean, Comm. Ann. Math. 33 (1932), 413-421. Google Scholar

[2] A. Alotaibi, A. M. Alroqi, Statistical convergence in a paranormed space, J. Inequal. Appl. 2012, 2012:39, 6 pp. Google Scholar

[3] M. Altınok, B. Inan, M. Ku ̈ ̧cu ̈kaslan, On deferred statistical convergence of sequences of sets in metric space, TJMCS, Article ID 20150050, (2015), 9 pages. Google Scholar

[4] M. Candan, A. Gu ̈ne ̧s, Paranormed sequence space of non-absolute type founded using generalized difference matrix, Proc. Nat. Acad. Sci. India Sect. A, 85 2 (2015), 269–276. Google Scholar

[5] M. Candan, A new perspective on paranormed Riesz sequence space of non-absolute type, Glob. J. Math. Anal., 3 4 (2015), 150–163. Google Scholar

[6] J. S. Connor, The statistical and strong p-Ces aro of sequences, Analysis, 8 (1988), 47-63. Google Scholar

[7] S. Ercan, Y. Altin, M. Et, V. K. Bhardwaj, On deferred weak statistical convergence, J. Anal., (2020), https://doi.org/10.1007/s41478-020-00221-5. Google Scholar

[8] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244. Google Scholar

[9] J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313. Google Scholar

[10] M. Ilkhan, E. E. Kara, A new type of statistical Cauchy sequence and its relation to Bourbaki completeness, Cogent Math. Stat., 5:1, 1487500 (2018), 9 pages. Google Scholar

[11] M. Ilkhan, E. E. Kara, On statistical convergence in quasi-metric spaces, Demonstr. Math. 52 (1) (2019), 225–236. Google Scholar

[12] C. Ko ̧sar, M. Ku ̈ ̧cu ̈kaslan, M. Et, On asymptotically deferred statistical equivalence of sequences, Filomat, 31:16 (2017), 5139–5150. Google Scholar

[13] M. Ku ̈ ̧cu ̈kaslan, M. Yılmaztu ̈rk,On deferred statistical convergence of sequences, Kyungpook Math. J., no. 2, 56 (2016), 357–366. Google Scholar

[14] I. J. Maddox, Elements of Functional Analysis, Cambridge at the University Press, (1970). Google Scholar

[15] A. Mohammed, M. Mursaleen, λ -statistical convergence in paranormed space, Abstr. Appl. Anal. 2013, Art. ID 264520, 5 pp. Google Scholar

[16] H. Roopaei, T. Yaying, Quasi-Ces`aro matrix and associated sequence space, Turkish J. Math. 45 (1) (2021), 153–166. Google Scholar

[17] T. Sal at, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139- 150. Google Scholar

[18] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–74. Google Scholar

[19] F. Temizsu, M. Et, M. C ̧ınar, ∆m-deferred statistical convergence of order α, Filomat 30 3 (2016), 667–673. Google Scholar

[20] Google Scholar

[21] N. Turan, E. E. Kara, M. Ilkhan, Quasi statistical convergence in cone metric spaces, Facta Univ. Ser. Math. Inform. 33 (4) (2018), 613–626. Google Scholar

[22] T. Yaying, B. Hazarika, Lacunary arithmetic statistical convergence, Natl. Acad. Sci. Lett., 43 (2020), 547–551. Google Scholar

[23] T. Yaying, B. Hazarika, M. Mursaleen, On sequence space defined by the domain of q-Ces`aro matrix in lp space and the associated operator ideal, J. Math. Anal. Appl. 493 (1) (2021), 124453. Google Scholar

[24] T. Yaying, Paranormed Riesz difference sequence spaces of fractional order, Kragujevac J. Math. 46 (2) (2022), 175–191. Google Scholar

[25] T. Yaying, B. Hazarika, A. Esi, Geometric properties and compact operators on fractional Riesz difference spaces, Kragujevac J. Math. 47 (4) (2023), 545–566. Google Scholar

[26] M. Yılmaztu ̈rk, M. Ku ̈c ̧u ̈kaslan, On strongly defereed Ces`aro summability and deferred statistical convergence of the sequences, Bitlis Eren Univ. J. Sci. Technology 3 (2011), 22–25. Google Scholar

[27] A. Zygmund, Trigonometrical Series, Monogr. Mat. 5. Warszawa-Lwow 1935. Google Scholar