On Asymptotically I-lacunary Statistical Equivalent Functions of Order $\alpha$

Abstract

The aim of this paper is to provide a new approach to some well known summability methods. We first define asymptotically ${\rm I}$-statistical equivalent functions of order $\alpha $, asymptotically ${\rm I} _{\theta} $-statistical equivalent functions of order $\alpha$ and strongly ${\rm I}$-lacunary equivalent functions of order $\alpha$ by taking two nonnegative real-valued Lebesgue measurable functions $x(t)$ and $y(t)$ in the interval $(1,\infty)$ instead of sequences and later we investigate their relationship.