Abstract
The main aim of the present article is to introduce some new ∇-conformable dynamic inequalities of Hardy type on time scales. We present and prove several results using chain rule and Fubini’s theorem on time scales. Our results generalize, complement, and extend existing results in the literature. Many special cases of the proposed results, such as new conformable fractional h-sum inequalities, new conformable fractional q-sum inequalities, and new classical conformable fractional integral inequalities, are obtained and analyzed.
Highlights
1 Introduction Fractional calculus theory has an important role in the mathematical analysis and applications
The corresponding fractional derivative is obtained by a composition of fractional integral with integer order derivative
Study on fractional dynamic equations is very widespread around the world and is useful in pure and applied mathematics, physics, engineering, biology, economics, etc
Summary
Fractional calculus theory has an important role in the mathematical analysis and applications. Fractional calculus (FC), the theory of integrals and derivatives of noninteger order, is a field of research with a history dating back to Abel, Riemann, and Liouville (see [33] for a historical summary). Study on fractional dynamic equations is very widespread around the world and is useful in pure and applied mathematics, physics, engineering, biology, economics, etc. They use an integral in its formulation, especially Cauchy’s integral formula with some modifications. Riemann–Liouville and Caputo fractional derivatives do not satisfy the nonlinear derivative rules as product, El-Deeb et al Journal of Inequalities and Applications (2021) 2021:192 quotient, and chain rules. The mean value theorem and Rolle’s theorem are not formulated using the definitions of Riemann–Liouville and Caputo fractional derivatives
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