Comparison of the remainders of the Simpson quadrature formula and the quadrature formula for three-point rational interpolants

A quadrature formula with positive coefficients is constructed with the use of three nodes $a$, $b$, and $c=(a+b)/2$ and rational interpolants of the form $\rho (x)= \alpha +\beta (x-c)+\gamma/(x-g)$ with a pole $g$ determined by nodes outside the integration interval $[a,b]$. The error of the constructed formula is smaller than the error of the corresponding Simpson quadrature formula if the integrand $f(x)$ has a continuous derivative $f^{(4)}(x)$ on the interval $[a,b]$ and the inequality $f^{(4)}(x) f^{\prime\prime}(x)>0$ is satisfied.