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Some Grüss type inequalities in inner product spaces. (English) Zbl 1063.26012

The follwing refinement of the Grüss inequality is proved. Let (H,·,·) be a real or complex inner product space and eH,e=1· If φ,γ,Φ,Γ are real or complex numbers and x,y are vectors in H such that

ReΦe-x,x-φe0andReΓe-y,y-γe0

hold, then we have the inequality

|x,y-x,ee,y|1 4|Φ-φ|·|Γ-γ|-[ReΦe-x,x-φe] 1/2 [ReΓe-y,y-γe] 1/2 ·


MSC:
26D15Inequalities for sums, series and integrals of real functions
46C05Hilbert and pre-Hilbert spaces: geometry and topology