A new factor theorem for generalized absolute Riesz summability. (English) Zbl 1446.40005
Summary: The aim of this paper is to consider an absolute summability method and generalize a theorem concerning \(\vert\bar{N},p_n\vert_k\) summability of infinite series [H. Bor, Indian J. Pure Appl. Math. 18, 330–336 (1987; Zbl 0631.40004)] to \(\varphi$-$\vert{\bar{N},p_n;\delta}\vert_k\) summability of infinite series by using an almost increasing sequence. Furthermore, it is explained that a well-known result [loc.cit.]dealing with \(\vert\bar{N},p_n\vert_k\) summability is obtained when this generalization is restricted under special conditions.
MSC:
| 40D15 | Convergence factors and summability factors |
| 40F05 | Absolute and strong summability |
| 40G99 | Special methods of summability |
| 26D15 | Inequalities for sums, series and integrals |
Keywords:
summability factors; almost increasing sequence; infinite series; Hölder inequality; Minkowski inequalityCitations:
Zbl 0631.40004References:
| [1] | Bari N. K., Stečkin S. B. Best approximations and differential properties of two conjugate functions. Trudy Moskov. Mat. Obšč. 1956, 5 , 483-522. (in Russian) · Zbl 0072.05702 |
| [2] | Bor H. On two summability methods. Math. Proc. Camb. Philos. Soc. 1985, 97 , 147-149. · Zbl 0554.40008 |
| [3] | Bor H. A note on \({\mid{\bar{N},p_n}\mid}_k\) summability factors of infinite series. Indian J. Pure Appl. Math. 1987, 18 , 330-336. · Zbl 0631.40004 |
| [4] | Bor H. On local property of \({\mid{\bar{N},p_n;\delta}\mid}_k\) summability of factored Fourier series. J. Math. Anal. Appl. 1993, 179 (2), 646-649. · Zbl 0797.42005 |
| [5] | Bor H., Seyhan H. On almost increasing sequences and its applications. Indian J. Pure Appl. Math. 1999, 30 , 1041-1046. · Zbl 0942.40001 |
| [6] | Bor H., Özarslan H. S. On absolute Riesz summability factors. J. Math. Anal. Appl. 2000, 246 , 657-663. · Zbl 0960.40004 |
| [7] | Bor H., Özarslan H. S. A note on absolute summability factors. Adv. Stud. Contemp. Math. (Kyungshang) 2003, 6 (1), 1-11. · Zbl 1036.40002 |
| [8] | Hardy G. H. Divergent Series. Oxford University Press, Oxford, 1949. · Zbl 0032.05801 |
| [9] | Karakaş A. A note on absolute summability method involving almost increasing and \(\delta \)-quasi-monotone sequences. Int. J. Math. Comput. Sci. 2018, 13 (1), 73-81. · Zbl 1418.40005 |
| [10] | Kartal B. On generalized absolute Riesz summability method. Commun. Math. Appl. 2017, 8 (3), 359-364. |
| [11] | Mazhar S. M. A note on absolute summability factors. Bull. Inst. Math. Acad. Sinica 1997, 25 , 233-242. · Zbl 0885.40004 |
| [12] | Mishra K. N., Srivastava R. S. L. On \(|\bar{N},p_n|\) summability factors of infinite series. Indian J. Pure Appl. Math. 1984, 15 , 651-656. · Zbl 0571.40008 |
| [13] | Özarslan H. S. On almost increasing sequences and its applications. Int. J. Math. Math. Sci. 2001, 25 , 293-298. · Zbl 0971.40004 |
| [14] | Özarslan H. S. A note on \(\left|\bar{N},p_n; \delta\right|_k\) summability factors. Indian J. Pure Appl. Math. 2002, 33 , 361-366. · Zbl 1004.40002 |
| [15] | Özarslan H. S. On \(\left|\bar{N},p_n; \delta\right|_k\) summability factors. Kyungpook Math. J. 2003, 43 , 107-112. · Zbl 1041.40002 |
| [16] | Seyhan H. On the local property of \(\varphi-{\mid{\bar{N},p_n;\delta}\mid}_k\) summability of factored Fourier series. Bull. Inst. Math. Acad. Sinica 1997, 25 , 311-316. · Zbl 0886.40004 |
| [17] | Seyhan H. A note on absolute summability factors. Far East J. Math. Sci. 1998, 6 (1), 157-162. · Zbl 1038.40004 |
| [18] | Seyhan H. On the absolute summability factors of type (A,B). Tamkang J. Math. 1999, 30 (1), 59-62. · Zbl 0986.40003 |
| [19] | Seyhan H. |
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