Abstract
We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as well as fractional integrals of a function with respect to another function. Finally, we apply our main result to multidimensional settings to obtain new results involving mixed Riemann-Liouville fractional integrals.
Highlights
Let us recall some facts about fractional derivatives needed in the sequel, for more details see, for example, 1, 2
By Cm a, b, we denote the space of all functions on a, b which have continuous derivatives up to order m, and AC a, b is the space of all absolutely continuous functions on a, b
We start with the definition of the Riemann-Liouville fractional integrals, see 3
Summary
Let us recall some facts about fractional derivatives needed in the sequel, for more details see, for example, 1, 2. By Cm a, b , we denote the space of all functions on a, b which have continuous derivatives up to order m, and AC a, b is the space of all absolutely continuous functions on a, b. By L1 a, b , we denote the space of all functions integrable on the interval a, b , and by L∞ a, b the set of all functions measurable and essentially bounded on a, b. We start with the definition of the Riemann-Liouville fractional integrals, see 3. The Riemann-Liouville fractional integrals Iaα f and Ibα− f of order α > 0 are defined by
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
