ABSTRACT
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.
