An inequality of Ostrowski type for twice differentiable mappings in terms of the \(L_p\) norm and applications (Q2739282)

The authors prove the following inequality of the Ostrowski type for twice differentiable mappings. Let \(g:[a,b]\rightarrow \mathbb{R}\) be a mapping whose first derivative is absolutely continuous on \([a,b]\). If we assume that the second derivative \(g''\in L_{p}(a,b)\), \(p\geq 1\), then we have the inequality NEWLINE\[NEWLINE \left|\int_{a}^{b}g(t) dt-\frac{1}{2}\left[ g(x)+\frac{g(a)+g(b)}{2}\right] (b-a)+(b-a)\left( x-\frac{a+b}{2}\right) g'(x)\right|\leq K_{p}(a,b,x)\cdot \left\|g''\right\|_{p}, NEWLINE\]NEWLINE for all \(x\in [a,b]\) , where NEWLINE\[NEWLINE \left\|g''\right\|_{p}=\left( \int_{a}^{b}\left|g''(t)\right|^{p} dt\right) ^{\frac{1}{p}} , NEWLINE\]NEWLINE NEWLINE\[NEWLINE K_{1}(a,b,x)=\frac{(b-a)^{2}}{8},\quad K_{2}(a,b,x)=\frac{(b-a)^{\frac{1}{2}}}{ 4}\left[ \left( x-\frac{a+b}{2}\right) ^{4}+\frac{(b-a)^{4}}{240}\right]^{ \frac{1}{2}},NEWLINE\]NEWLINE but the expression of \(K_{p}(a,b,x)\), for an arbitrary \(p\), is too complicated to be detailed here. Some applications to special means and numerical integration are also given.