Abstract
We consider the SteffensenâHayashi inequality and remainder identity for V-fractional differentiable functions involving the six parameters truncated MittagâLeffler function and the Gamma function. In view of these, we obtain some integral inequalities of Steffensen, HermiteâHadamard, Chebyshev, Ostrowski, and GrĂŒss type to the V-fractional calculus.
Highlights
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One useful and important branch of science which involves derivatives and integrals taken to fractional orders is fractional calculus, in general [1,2,3,4]
Let us recall the six truncated MittagâLeffler function and the truncated V -fractional derivative which will be used in the sequel
Summary
Sousa and Oliveira [6] defined M-fractional derivative via MittagâLeffler function of one parameter [7]. Sousa and Oliveira [8] introduced the V -fractional derivative involving the six parameters truncated MittagâLeffler function and the Gamma function. Let us recall the six truncated MittagâLeffler function and the truncated V -fractional derivative which will be used in the sequel. Six parameters truncated MittagâLeffler function is defined by:. (ii) From (1), we can obtain directly by determining some parameters to be 1, some particular cases regarding the following truncated MittagâLeffler functions:. The truncated V -fractional derivative of f of order ÎŒ > 0, is given as:. Several results similar to the results found in the classical calculus are obtained from the truncated V -fractional derivative using the six parameters truncated MittagâLeffler function.
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