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On statistical convergence and strong Cesàro convergence by moduli

2019; Springer Science+Business Media; Volume: 2019; Issue: 1 Linguagem: Inglês

10.1186/s13660-019-2252-y

1029-242X

Fernando León-Saavedra, María del Carmen Listán-García, Francisco Javier Fernández, María del Pilar Romero de la Rosa,

Fuzzy and Soft Set Theory

Abstract In this paper we will establish a result by Connor, Khan and Orhan (Analysis 8:47–63, 1988; Publ. Math. (Debr.) 76:77–88, 2010) in the framework of the statistical convergence and the strong Cesàro convergence defined by a modulus function f . Namely, for every modulus function f , we will prove that a f -strongly Cesàro convergent sequence is always f -statistically convergent and uniformly integrable. The converse of this result is not true even for bounded sequences. We will characterize analytically the modulus functions f for which the converse is true. We will prove that these modulus functions are those for which the statistically convergent sequences are f -statistically convergent, that is, we show that Connor–Khan–Orhan’s result is sharp in this sense.