chapter  4
46 Pages

Gru¨ss Type Inequalities and Related Results

Let f, g : [a, b]→ R be integrable on [a, b] and satisfy

φ ≤ f (x) ≤ Φ, γ ≤ g (x) ≤ Γ for all x ∈ [a, b] . (4.1)

Then we have the inequality

|T (f, g)| (4.2)

:=

∣∣∣∣∣ 1b− a ∫ b a

f (x) g (x) dx− 1 b− a

f (x) dx · 1 b− a

g (x) dx

∣∣∣∣∣ ≤ 1

4 (Φ− φ) (Γ− γ) ,

where T (f, g) is known as the Cˇebysˇev functional.