Abstract
We develop the Benkhettou–Hassani–Torres fractional (noninteger order) calculus on timescales by proving two chain rules for the alpha -fractional derivative and five inequalities for the alpha -fractional integral. The results coincide with well-known classical results when the operators are of (integer) order alpha = 1 and the timescale coincides with the set of real numbers.
Highlights
The study of fractional calculus on timescales is a subject of strong current interest [1,2,3,4]
We develop the BHT timescale fractional calculus initiated in [5]
We briefly recall the necessary notions from the BHT fractional calculus [5]: fractional differentiation and fractional integration on timescales
Summary
The study of fractional (noninteger order) calculus on timescales is a subject of strong current interest [1,2,3,4]. We develop the BHT timescale fractional calculus initiated in [5]. We prove two different chain rules for the fractional derivative Tα (Theorems 3.1 and 3.3) and several inequalities for the α-fractional integral: Hölder’s inequality (Theorem 3.4), Cauchy–Schwarz’s inequality (Theorem 3.5), Minkowski’s inequality (Theorem 3.7), generalized Jensen’s fractional inequality (Theorem 3.8) and a weighted fractional Hermite– Hadamard inequality on timescales (Theorem 3.9). 2, we recall the basics of the the BHT fractional calculus. Our results are formulated and proved in Sect.
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