In this section, we define the notion of strong summability by a modulus function and establish its relation with statistical convergence in a paranormed space.
Recall in [25] that an Orlicz function M is a continuous, convex, nondecreasing function defined for x > 0 such that M ( 0 ) = 0 and M ( x ) > 0. If the convexity of an Orlicz function is replaced by M ( x + y ) ≤ M ( x ) + M ( y ), then this function is called the modulus function and characterized by Ruckle [26].
After then some sequence spaces defined by a modulus function were introduced and studied by Bilgin [12], Pehlivan and Fisher [13], Waszak [14], Bhardwaj [15], Altın [16], and many others.
By definition of modulus function (iii) and (ii), we have, for (| x_{k}| > delta), fbigl(| x_{k}|bigr) leq bigl(1 + bigl[| x_{k}|/delta bigr] bigr)f(1) leq 2f(1)| x_{k}|/delta.
By definition of modulus function (iii) and (ii), we have frac{1}{n^{alpha}}sum_{k=1}^{n}fbigl(| x_{k}|bigr) leq frac{1}{n^{alpha}}sum_{k=1}^{n}fbigl(| x_{k} - l|bigr) + fbigl(| l |bigr)frac{1}{n^{alpha}}sum _{k=1}^{n}1, and since (alphageq1) and (x in w_{alpha}^{f}), we have (x in w_{alpha,infty}^{f}), which completes the proof.
then this function is called modulus function, introduced and investigated by Nakano [9] and followed by Ruckle [10], Maddox [11], and many others.
In this article, we define and study the notion of convergence, statistical convergence, statistical Cauchy, and strong summability by a modulus function in a paranormed space.
If convexity of Orlicz function, is replaced by, then this function is called Modulus function, which was presented and discussed by Ruckle [13] and Maddox [14].
We now consider some new kind of asymptotically equivalent sequences defined by ideals, lacunary sequences and a modulus function.
Let φ and f be a given φ-function and a modulus function, respectively.
