*Article* **Weighted Fractional Iyengar Type Inequalities in the Caputo Direction**

#### **George A. Anastassiou**

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA; ganastss@memphis.edu

Received: 8 October 2019; Accepted: 14 November 2019; Published: 16 November 2019

**Abstract:** Here we present weighted fractional Iyengar type inequalities with respect to *Lp* norms, with 1 ≤ *p* ≤ ∞. Our employed fractional calculus is of Caputo type defined with respect to another function. Our results provide quantitative estimates for the approximation of the Lebesgue–Stieljes integral of a function, based on its values over a finite set of points including at the endpoints of its interval of definition. Our method relies on the right and left generalized fractional Taylor's formulae. The iterated generalized fractional derivatives case is also studied. We give applications at the end.

**Keywords:** Iyengar inequality; right and left generalized fractional derivatives; iterated generalized fractional derivatives; generalized fractional Taylor's formulae

**MSC:** 26A33; 26D10; 26D15
