Taylor-Widder representation formulae and Ostrowski, Grüss, integral means and Csiszar type inequalities (Q2460558)

For \(f,h,u_{0},u_{1},\dots,u_{n}\in C^{n+1}[a,b]\), \(n\geq 0,\) consider \( L_{0}f(x):=f(x),\) \[ L_{i}f(x):= \frac{W[u_{0}(x),u_{1}(x),\dots,u_{i-1}(x),f(x)]} {W_{i-1}(x)}, \quad i=1,\dots,n+1;\;\forall x\in [ a,b], \] \(g_{0}(x,t):=\frac{u_{0}(x)}{u_{0}(t)}\), \(\forall x,t\in [ a,b]\) and \[ g_{i}(x,t):=\frac{1}{W_{i}(t)}\left| \begin{matrix} u_{0}(t) & u_{1}(t) & \cdots & u_{i}(t) \\ u_{0}^{\prime }(t) & u_{1}^{\prime }(t) & \cdots & u_{i}^{\prime }(t) \\ \vdots & \vdots & & \vdots \\ u_{0}^{(i-1)}(t) & u_{1}^{(i-1)}(t) & \cdots & u_{i}^{(i-1)}(t) \\ u_{0}(x) & u_{1}(x) & \cdots & u_{i}(x) \end{matrix}\right| , \quad i=1,2,\dots,n, \] where \(W[u_{0}(x),u_{1}(x),\dots,u_{i-1}(x),f(x)]\) denotes the Wronskian of the functions \(u_{0},u_{1},\dots,u_{i-1}\), \(f\) and \( W_{i}(x)=W[u_{0}(x),u_{1}(x),\dots,u_{i}(x)]\) is assumed strictly positive for \(i=0,1,\dots,n\) and \(x\in [ a,b].\) Using the Taylor-Widder representation formula, the author proves the following Grüss type inequality: \[ \begin{multlined} \left| \frac{1}{b-a}\int_{a}^{b}f(x)h(x)\,dx- \left( \frac{1}{b-a} \int_{a}^{b}f(x)dx\right) \left( \frac{1}{b-a}\int_{a}^{b}f(x)\,dx\right) \right.\\ \left. -\frac{1}{2(b-a)^{2}}\sum_{i=1}^{n}\int_{a}^{b} \int_{a}^{b} (h(x)L_{i}f(y)+ f(x)L_{i}h(y))g_{i}(x,y)\,dy\,dx\right|\\ \leq \frac{1}{2(b-a)^{2}} \sum_{i=1}^{n} \int_{a}^{b} \int_{a}^{b} (| h(x)| \| L_{n+1}f\| _{\infty }+| f(x)| \| L_{n+1}h)\| _\infty)| g_{n}(x,t)| | k(t,x)| \,dt\,dx, \end{multlined} \] where \[ k(t,x):=\begin{cases} t-a, &a\leq t\leq x\leq b\\ t-b, &a\leq x<t\leq b. \end{cases} \] Ostrowski integral means and Csiszar type inequalities are also proved.