A new simple proof for an inequality of Cebyshev type

We give here a simple proof of a well-known integral version of Cebyshev inequality. Using the same method, we give a lower bound in the case of increasing functions and then in the case of convex functions. We also establish a result at limit which shows that the constant 1/12 is sharp, in the sense that it cannot be replaced by a smaller one. Subject Classification: 26D15. It is mentioned in [2, pp. 297] the following inequality of Cebyshev type: Theorem 1. Let f, g : [a, b] → R be derivable functions, with bounded derivatives on [a, b] ⊆ R. Then ∣∣∣∣ 1 b− a ∫ b a f(x)g(x) dx− 1 b− a ∫ b a f(x) dx · 1 b− a ∫ b a g(x) dx ∣∣∣∣ ≤ ≤ (b − a) 2 12 · ||f ′|| · ||g′|| , where ||f ′|| = sup x∈[a,b] |f ′(x)| , ||g′|| = sup x∈[a,b] |g′(x)| . The constant 1/12 is the best possible one in the sense that it cannot be replaced by a smaller one.