On statistical convergence and strong Cesaro 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 Cesaro convergence defined by a modulus function f. Namely, for every modulus function f, we will prove that a f-strongly Cesaro 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.