The extension of Montgomery identity via Fink identity with applications (Q2569961)

Let \(w:[a,b]\rightarrow [ 0,\infty )\) be some probability density function, that is, an integrable function satisfying \(\int_{a}^{b}w(t)dt=1.\) Denote \(W(t)=\int_{a}^{t}w(x)dx\) for \(t\in [ a,b]\), \[ P_{w}(x,t)=\begin{cases} W(t), & a\leq t\leq x, \\ W(t)-1, & x<t\leq b \end{cases} \quad\text{and}\quad k(t,x)=\left\{ \begin{matrix} t-a,\quad a\leq t\leq x\leq b, \\ t-b,\quad a\leq x<t\leq b. \end{matrix} \right. \] Let also \(f:[a,b]\rightarrow \mathbb{R}\) be such that \(f^{(n-1)}\) is an absolutely continuous function on \([a,b]\) for some \(n\geq 2\) and denote \[ F_{j}(x)=\frac{n-j}{j!}\frac{f^{(j-1)}(a)(x-a)^{j}-f^{(j-1)}(b)(x-b)^{j}}{b-a },\quad 1\leq j<n. \] After giving a new extension of the weighted Montgomery identity, the authors prove the following Ostrowski-type inequality: if \(\left| f^{(n)}\right| ^{p}\) is R-integrable for some \(1\leq p\leq \infty ,\) then \[ \begin{multlined} \left| f(x)-\int_{a}^{b}w(t)f(t)dt+\sum_{j=1}^{n-1}\left[ F_{j}(x)-\int_{a}^{b}w(t)F_{j}(t)dt\right] \right| \\ \leq \frac{\left\| f^{(n)}\right\| _{p}}{(n-2)!(b-a)}\left( \int_{a}^{b}\left| \int_{a}^{b}P_{w}(x,t)(t-y)^{n-2}k(y,t)dt\right| ^{q}dy\right) ^{1/q}\end{multlined} \] holds for \(x\in [ a,b],\) where \(1\leq q\leq \infty \) is such that \( 1/p+1/q=1.\)