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In this paper, following a very recent and new approach of Aizpuru et al. (Quaest. Math. 37:525-530, 2014), we further generalize a concept of α-density to that of \(f_{\alpha}\)-density, where f is an unbounded modulus and \(0 < \alpha\leq1\). As a consequence, we obtain a new nonmatrix convergence method, namely f-statistical convergence of order α or \(S_{\alpha}^{f}\)-convergence, which is intermediate between the ordinary convergence and the statistical convergence of order α. We also introduce a new concept of strong Cesàro summability of order α with respect to a modulus function f, and finally we investigate the relationship between the set \(S_{\alpha}^{f}\) of all f-statistically convergent sequences of order α and the set \(w_{\alpha}^{f}\) of all strongly Cesàro summable sequences of order α with respect to f.

f-Statistical convergence of order α and strong Cesàro summability of order α with respect to a modulus