Grüss' inequality for positive linear functionals
Let X be an arbitrary nonvoid set and let F(X) denote the set of all real-valued functions on X. It is proved (Grüss' inequality) that if \(A:F(X)\mapsto {\mathbb{R}}\) is a positive linear functional and \(f,g\in F(X)\) satisfy \(m_ 1\leq f\leq M_ 1\) and \(m_ 2\leq g\leq M_ 2,\) then \[ | A(e_ 0)A(fg)-A(f)A(g)| \leq (A^ 2(e_ 0)/4)(M_ 1-m_ 1)(M_ 2-m_ 2). \] Here \(e_ 0(x)=1\). If \(A(f)=\int ^{b}_{a}p(x)f(x)dx,\) where p is a positive integrable function on \(X=[a,b]\) and A is defined only for integrable functions in F(X), then \(A^ 2(e_ 0)/4\) is the best possible constant. If A is the discrete functional \(f\mapsto (\sum ^{n}_{i=1}p_ if(x_ i))/P_ n,\) \(P_ n=\sum ^{n}_{i=1}p_ i,\) \(p_ i\geq 0,\) then the above bound in Grüss' inequality may be replaced by \((\sum _{i\in S}p_ i)(P_ n- \sum _{i\in S}p_ i)(M_ 1-m_ 1)(M_ 2-m_ 2),\) where S is the subset of \(\{1,...,n\}\) which minimizes the expression \(| \sum _{i\in S}p_ i-P_ n/2|.\) This is also the best possible bound. Finally, three applications are given.
scientific article; zbMATH DE number 4004518