Identities for the Chebyshev functional involving derivatives of arbitrary order and applications
Let \(\mu \) be a normalized (signed) measure on \([0,1]\) and \(L^{1}(\mu )\) be the space of integrable functions with respect to \(\mu .\) For \(f,g,fg\in L^{1}(\mu ),\) the Chebyshev functional is defined by \[ T(f,g;\mu )=\int_{0}^{1}fgd\mu -\int_{0}^{1}fd\mu \int_{0}^{1}gd\mu . \] The authors prove a family of identities, whose simplest element is the following: if \(f\) and \(g\) are such that \(f^{(n)}\) and \(g^{(n)}\) are integrable on \([0,1],\) then \[ T(f,g;\mu )=\sum_{i=1}^{n-1}(-1)^{i+1}\int_{0}^{1}\left[ R_{i}L_{i+1}+L_{i}R_{i+1}\right] g^{(i)}f^{(i)} \] \[ +(-1)^{n+1}\int_{0}^{1}\int_{0}^{1}k_{n,n}(x,t)g^{(n)}(t)f^{(n)}(x)dtdx, \] where \[ L_{1}(x)=\int_{0}^{x}d\mu ,\quad L_{n}(x)=\int_{0}^{x}L_{n-1}(x)dx,\;n\geq 2, \] \[ R_{1}(x)=\int_{x}^{1}d\mu ,\quad R_{n}(x)=\int_{x}^{1}R_{n-1}(x)dx,\;n\geq 2. \] The identities are very useful in establishing bounds for the Chebyshev functional.
scientific article; zbMATH DE number 5016114