Hermite-Hadamard Inequality and Quadrature Formulae

We now turn to the question of how the Hermite-Hadamard inequality arises in various formulae in mathematics, specifically quadrature formulae which allow for numerical approximation of definite integrals.

For f \in C^{2}([a,b],\mathbb{R}), we can integrate by parts to obtain

\begin{displaystyle}\begin{array}{lcl}\frac{1}{b-a}\int_{a}^{b}\left[f(x)-\frac{f(a)+f(b)}{2}\right]dx&=&\frac{1}{b-a}\left[x\left(f(x)-\frac{f(a)+f(b)}{2}\right)|_{a}^{b}-\int_{a}^{b}f'(x)xdx\right]\\&=&\frac{1}{b-a}\left[b\frac{f(b)-f(a)}{2}+a\frac{f(b)-f(a)}{2}-\int_{a}^{b}f'(x)xdx\right]\\&=&\frac{1}{b-a}\left[\int_{a}^{b}f'(x)\left(\frac{a+b}{2}-x\right)dx\right]\\&=&\frac{1}{b-a}\left[f'(x)\left(x\frac{a+b}{2}-\frac{1}{2}x^{2}\right)|_{a}^{b}-\int_{a}^{b}f''(x)\left(x\frac{a+b}{2}-\frac{1}{2}x^{2}\right)dx\right]\\&=&\frac{1}{b-a}\left[f'(b)\frac{ab}{2}-f'(a)\frac{ab}{2}-\int_{a}^{b}f''(x)\left(x\frac{a+b}{2}-\frac{1}{2}x^{2}\right)dx\right]\\&=&\frac{1}{b-a}\left[\int_{a}^{b}f''(x)\left(\frac{ab}{2}-x\frac{a+b}{2}+\frac{1}{2}x^{2}\right)dx\right]\\&=&\frac{1}{b-a}\int_{a}^{b}f''(x)\frac{(x-a)(x-b)}{2}dx\end{array}\end{displaystyle}

If f is in addition convex (i.e. f''\geq 0), then the the preceding quadrature formula gives us the right Hermite-Hadamard inequality.

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