On Conformable Fractional Riesz Bounded \(\user2{ p} -\) Variation of Order \(- \user2{ \alpha }\)

In this paper, inspired by the concept of conformable fractional (CF) derivative of order \(\alpha\) with the definitions of Riesz bounded \(p -\) variation and Lipschitz continuous function, we introduce the new definitions of CF Riesz bounded variation and CF Lipschitz continuous function, respectively. Furthermore, we give some of its important properties and the relationships among those functions. In particular, the following chain of inclusions holds: \(\rm{\mathbb{D}}^{\alpha} \left[ {a,b} \right] \subset \rm{{\mathbb{L}}{\mathbb{I}}{\mathbb{P}}}^{\alpha} \left[ {a,b} \right] \subset \rm{{\mathbb{B}}{\mathbb{V}}}^{\alpha,p} \left[ {a,b} \right] \subset \rm{{\mathbb{B}}{\mathbb{V}}}^{\alpha,q} \left[ {a,b} \right].\) where \(\alpha \in \left( {0,} \right.\left. 1 \right]\) and \(p,q \in \left( {1,\infty } \right)\) where \(q < p\) .