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inequality; norm; summability matrix; Hausdorff matrix; Nörlund matrix; weighted mean matrix; weighted sequence space and Lorentz sequence space
In this paper we consider the problem of finding upper bounds of certain matrix operators such as Hausdorff, Nörlund matrix, weighted mean and summability on sequence spaces $l_p(w)$ and Lorentz sequence spaces $d(w,p)$, which was recently considered in [9] and [10] and similarly to [14] by Josip Pecaric, Ivan Peric and Rajko Roki. Also, this study is an extension of some works by G. Bennett on $l_p$ spaces, see [1] and [2].
[1] G. Bennett: Factorizing the classical inequalities. Mem. Amer. Math. Soc. 576 (1996), 1–130. MR 1317938 | Zbl 0857.26009
[2] G. Bennett: Inequalities complimentary to Hardy. Quart. J. Math. Oxford (2) 49 (1998), 395–432. MR 1652236 | Zbl 0929.26013
[3] D. Borwein and F. P. Cass: Nörlund matrices as bounded operators on $l_p$. Arch. Math. 42 (1984), 464–469. DOI 10.1007/BF01190697 | MR 0756700
[4] D. Borwein: Nörlund operators on $l_p$. Canada. Math. Bull. 36 (1993), 8–14. DOI 10.4153/CMB-1993-002-x | MR 1205888
[5] G. H. Hardy: An inequality for Hausdorff means. J. London Math. Soc. 18 (1943), 46–50. MR 0008854 | Zbl 0061.12704
[6] G. H. Hardy: Divergent Series. 2nd edition, American Mathematical Society, 2000.
[7] G. H. Hardy and J. E. Littlewood: A maximal theorem with function-theoretic. Acta Math. 54 (1930), 81–116. DOI 10.1007/BF02547518 | MR 1555303
[8] G. H. Hardy, J. E. Littlewood and G. Polya: Inequalities. 2nd edition, Cambridge University press, Cambridge, 2001.
[9] G. J. O. Jameson and R. Lashkaripour: Lower bounds of operators on weighted $l_p$ spaces and Lorentz sequence spaces. Glasgow Math. J. 42 (2000), 211–223. DOI 10.1017/S0017089500020061 | MR 1763740
[10] G. J. O. Jameson and R. Lashkaripour: Norms of certain operators on weighted $l_p$ spaces and Lorentz sequence spaces. J. Inequalities in Pure and Applied Mathematics, 3, Issue 1, Article 6 (2002). MR 1888921
[11] R. Lashkaripour: Lower bounds and norms of operators on Lorentz sequence spaces. Doctoral dissertation (Lancaster, 1997).
[12] R. Lashkaripour: Transpose of the Weighted Mean operators on Weighted Sequence Spaces. WSEAS Transaction on Mathematics, Issue 4, 4 (2005), 380–385.
[13] R. Lashkaripour and D. Foroutannia: Lower Bounds for Matrices on Weighted Sequence Spaces. Journal of Sciences Islamic Republic of IRAN, 18 (2007), 49–56. MR 2499829
[14] J. Pecaric, I. Peric and R. Roki: On bounds for weighted norms for matrices and integral operators. Linear Algebra and Appl. 326 (2001), 121–135. MR 1815954
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