RELATIONSHIP OF UPPER BOX DIMENSION BETWEEN CONTINUOUS FRACTAL FUNCTIONS AND THEIR RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL

This paper considers the relationship of box dimension between a continuous fractal function and its Riemann-Liouville fractional integral. For an arbitrary fractal function f(x) it is proved that the upper box dimension of the graph of Riemann-Liouville fractional integral D(-nu)f(x) does not exceed the upper box dimension of f(x), i.e. dim over bar(B)Upsilon(D-nu f,I) <= dim over bar(B) Upsilon(f,I). This estimate shows that nu-order Riemann-Liouville fractional integral D(-nu)f(x) does not increase the fractal dimension of the integrand f(x), which means that Riemann-Liouville fractional integration does not decrease the smoothness at least that is obvious known result for classic integration. Our result partly answers fractal calculus conjecture in [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 2 (1995) 217-229] and [Y. S. Liang and W. Y. Su, Riemann-Liouville fractional calculus of one-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423-438].