# A fractional dimension-space theorem for Roll's and mean value theorems

### Abstract

In this paper, we briefly review some accomplished research in the mean value and Roll's theorems of the fractional calculus. Then, we present a mathematical space so-called Fractional Dimension Space (FDS). Using this space, a Roll's and mean value transformation is made, which transmits the classical Roll's and mean value theorems into the FDS'. The goal is finding a fractional order $0\leq \alpha<1$ by one of the well-known methods such as Riemann-Liouville, Caputo, Gr$\ddot{u}$nwald-Letnikov, Hadamard, and Weyl fractional derivatives satisfying the classical mean value and Roll's theorems. Finally, we give a theorem proving that there exists an $0\leq \alpha<1$ in such space through at least one of the aforementioned methods of differentiation. The two applications of the FDS are mentioned in this paper as well.