The Simpson's Inequality for r-Convex Mappings

Abstract

For an absolutely continuous mapping $f^ {\prime \prime \prime}:I\subseteq \mathbb{R}\rightarrow \mathbb{R} $, on $I^{\circ }$, where $a,b\in I$ with $a<b$. It is proved that, if $\left\vert {f^{(4)}}\right\vert $ is convex on $[a,b]$, then inequality \begin{align*} \label{Simp.ineq}\left\vert {\frac{1}{6}\left[ {f\left( a \right) + 4f\left( {\frac{{a + b}}{2}} \right) + f\left( b \right)} \right] - \frac{1}{b- a}\int_a^b {f\left( x \right)dx}}\right\vert \le \frac{{\left( {b - a} \right)}}{{241920}}\left( {23\left| {f^{\left( 4 \right)} \left( a \right)} \right| + 19\left| {f^{\left( 4 \right)} \left( b \right)} \right|} \right). \end{align*} holds.