Comparing two integral means for absolutely continuous mappings whose derivatives are in \(L_\infty[a,b]\) and applications (Q1609177)

Let \(f:[a,b]\rightarrow \mathbb{R}\) be an absolutely continuous mapping with \( f^{\prime }\in L_{\infty }[a,b]\) , i. e. \(\left\|f^{\prime }\right\|_{\infty }=ess \sup_{t\in [a,b]}\left|f^{\prime }(t)\right|<\infty .\) The authors prove that for \([c,d]\subset [a,b]\) is valid the following estimate: \[ \left|\frac{1}{b-a}\int_{a}^{b}f(t)dt-\frac{1}{d-c}\int_{c}^{d}f(t)dt \right|\leq \frac{1}{2}[(b-a)-(d-c)]\left\|f^{\prime }\right\|_{\infty }. \] This is a generalization of the classical Ostrowski inequality which may be deduced for \(c=x\in \left( a,b\right) \) and \(d\rightarrow x\) . Applications for probability density functions, special means, Jeffreys' divergence and continuous streams are given.