A generalization of Grüss's inequality in inner product spaces and applications (Q1306853)

The following generalization of the classical Grüss integral inequality is proved: Let \(X\) be a real or complex inner product space and \(e\in X\), \(\|e\|= 1\). If \(\varphi\), \(\gamma\), \(\Phi\), \(\Gamma\) are (real or complex) numbers and \(x,y\in X\) vectors such that \(\text{Re}(\Phi e-x,x- \varphi e)\geq 0\), and \(\text{Re}(\Gamma e- y,y-\gamma e)\geq 0\), then \(|(x,y)- (x,e)(e,y)|\leq{1\over 4}|\Phi- \varphi|\cdot|\Gamma- \gamma|\). The constant \(1/4\) is the best possible. Some applications of this result for positive linear functionals and integrals are given.