Iyengar-Hilfer fractional inequalities (Q2065507)

\textit{K. S. K. Iyengar} [Math. Stud. 6, 75--76 (1938; JFM 64.0209.02)] obtained the following classical inequality: if \(f\) is a differentiable function on the real interval \([a,b]\) and \(| f'(x)| \leq M,\) then \[\Bigl| \int^{b}_{a}f(x)dx-\frac{1}{2}(b-a)(f(a)+f(b))\Bigr| \leq \frac{M(b-a)^2}{4} -\frac{(f(b)-f(a))^2}{4M}.\] The above result is often referred to as Iyengar inequality in the literature. In this paper, the author presents Iyengar-type inequalities involving the \(\Psi\)-Hilfer and Hilfer left and right fractional derivatives with respect to norms \(\| . \|_p\), \(1 \leq p \leq \infty\) at the univariate and multivariate levels. Furthermore, the multivariate cases of radial and non-radial approaches over the shell and ball are also presented and discussed. An application of the results obtained for \(\Psi (x) = e^x\) is included.