On statistical convergence and strong Cesàro convergence by moduli

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.