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A new class of sequences related to the ${l}_{p}$ spaces defined by sequences of Orlicz functions. (English) Zbl 1236.46006

A new sequence space $m\left(M,\varphi ,q,{\Lambda }\right)$ is defined using an Orlicz function, seminorms and $\lambda$-sequences in order to generalize the space $m\left(\varphi \right)$, introduced and studied by W.L.C. Sargent in 1960. Later on, many authors defined several other spaces in comparison with the space $m\left(\varphi \right)$. First, the author investigates the linearity, examines solidity and monotonicity and gives some inclusion results involving the spaces $m\left(M,\varphi ,q,{\Lambda }\right)$. Then the relation between the space $m\left(M,\varphi ,q,{\Lambda }\right)$ and the space of ${S}_{\theta }^{0}\left(\varphi ,{\Lambda }\right)$-statistically convergent sequences is studied. In the last part, the relation between the space $m\left(M,\varphi ,q,{\Lambda }\right)$ and the space ${m}_{\theta }^{c}\left(M,\varphi ,q,{\Lambda }\right)$ of Cesàro convergence type sequences is given.

Reviewer’s remark: Certain properties like solid space, monotonicity of the space $m\left(M,\varphi ,q,{\Lambda }\right)$ are investigated without any specific aim and almost all results are generalized versions of other results already studied by others.

##### MSC:
 46A45 Sequence spaces 40A05 Convergence and divergence of series and sequences 40G05 Cesàro, Euler, Nörlund and Hausdorff methods
##### References:
 [1] [2] [3]