A New Estimation Of Box Dimension Of Riemann–Liouville Fractional Calculus Of Continuous Functions

This paper establishes a linear relationship between the order of the Riemann–Liouville fractional calculus and the exponent of the Hölder condition, whether the Hölder condition is global, local, or at a single point. We propose and prove a control inequality between the Hölder derivative (Hf(x,α) as defined in Proposition 12) of a continuous function and the Hölder derivative of the Riemann–Liouville fractional calculus of this function. In addition, this paper provides a more accurate estimation of the Box dimension of the graph of the Riemann–Liouville fractional integral of an arbitrary continuous function. More specifically, it establishes the result that whenever there is a continuous function whose graph has the upper Box dimension s with 1 < s ≤ 2, the graph of its Riemann–Liouville fractional integral of order ν, with 0 < ν < 1, has the upper Box dimension not greater than s − (s − 1)ν.