More efficient estimates via ℏ-discrete fractional calculus theory and applications

Abstract Discrete fractional calculus ( DFC ) is continuously spreading in the engineering practice, neural networks, chaotic maps, and image encryption, which is appropriately assumed for discrete-time modelling in continuum problems. First, we start with a novel discrete ℏ -proportional fractional sum defined on the time scale ℏ Z so as to give the premise to the more broad and complex structures, for example, the suitably accustomed transformations conjuring the property of observing the new chaotic behaviors of the logistic map. Here, we aim to present the novel discrete versions of Gruss and certain other associated variants by employing discrete ℏ -proportional fractional sums are established. Moreover, several novel consequences are recaptured by the ℏ -discrete fractional sums. The present study deals with the modification of Young, weighted-arithmetic and geometric mean formula by taking into account changes in the exponential function in the kernel represented by the parameters of the operator, varying delivery noted outcomes. In addition, two illustrative examples are apprehended to demonstrate the applicability and efficiency of the proposed technique.