Density by Moduli and Statistical Boundedness

Abstract

We have generalized the notion of statistical boundedness by introducing the concept of -statistical boundedness for scalar sequences where is an unbounded modulus. It is shown that bounded sequences are precisely those sequences which arestatistically bounded for every unbounded modulus . A decomposition theorem for -statistical convergence for vector valued sequences and a structure theorem for -statistical boundedness have also been established.

On Asymptotically f-statistical Equivalent Set Sequences in the Sense of Wijsman

“…In the year 2014, Aizpuru et al [1] defined a new concept of density with the help of an unbounded modulus function and, as a consequence, they obtained a new concept of non-matrix convergence, namely, f -statistical convergence, which is intermediate between the ordinary convergence and the statistical convergence and agrees with the statistical convergence when the modulus function is the identity mapping. Quite recently, Bhardwaj and Dhawan [3], and Bhardwaj et al [4], have introduced and studied the concepts of f -statistical convergence of order α and f -statistical boundedness, respectively, by using the approach of Aizpuru et al [1] (see also [5][6][7][8]).…”

Asymptotically deferred f-statistical equivalence of sequences

“…In 2014, Aizpuru et al [2] defined a new concept of density with the help of an unbounded modulus function and, as a consequence, they obtained a new concept of nonmatrix convergence, namely, f -statistical convergence, which is intermediate between the ordinary convergence and the statistical convergence and agrees with the statistical convergence when the modulus function is the identity mapping. Quite recently, Bhardwaj and Dhawan [4], and Bhardwaj et al [5], have introduced and studied the concepts of f -statistical convergence of order α and f -statistical boundedness, respectively, by using the approach of Aizpuru et al [2] (see also [6,7]).…”

Density by moduli and Wijsman lacunary statistical convergence of sequences of sets

“…This idea of replacing natural density with density by moduli has motivated us to look for some new generalizations of statistical convergence and consequently we have introduced and studied the concepts of f -statistical convergence of order α [34] and f -lacunary statistical convergence [35]. Using the notion of density by moduli Bhardwaj et al [36] have also introduced and studied the concept of f -statistical boundedness which is a generalization of statistical boundedness [38] and intermediate between the usual boundedness and the statistical boundedness.…”