Estimation of Integral Inequalities Using the Generalized Fractional Derivative Operator in the Hilfer Sense

With the great progress of fractional calculus, integral inequalities have been greatly enriched by fractional operators; users and researchers have formed a real-world phenomenon in the production of the evaluation process, which results in convexity. Monotonicity and inequality theory has a strong relationship, whichever we work on, and we can apply it to the other one due to the strong correlation produced between them, especially in the past few years. In this article, we introduce some estimations of left and right sides of the generalized Caputo fractional derivatives of a function for nth order differentiability via convex function, and related inequalities have been presented. Monotonicity and convexity of functions are used with some usual and straightforward inequalities. Moreover, we establish some new inequalities for C⌣ eby s⌣ ev and Gru¨ss type involving the generalized Caputo fractional derivative operators. The finding provides the theoretical basis and practical significance for the establishment of fractional calculus in convexity. It also introduces new ways of thinking and methods for innovative scientific research.