ABSTRACT
Proposition 2.2 (Jensen inequality). Let φ(x) be a convex function (so that φ′′(x) > 0) defined on the interval [a, b] ∈ R and xi , i = 1, . . . , n be some points in the interval: xi ∈ [a, b]. Then
αiφ(xi ) ≥ φ ( n∑
) (C.2)
where n∑
i=1 αi = 1 αi ∈ R+ (positive numbers). (C.3)
Proof of the Jensen inequality. It is clear that x˜ ≡ i=1 αi xi belongs to the interval [a, b]. Consider the Taylor series with the remainder term
φ(xi ) = φ(˜x)+ (xi − x˜)φ′(˜x)+ 12 (xi − x˜)2φ′′(ξi ) i = 1, . . . , n where ξi ∈ [a, b]. Since the function φ is convex, we have φ′′(x) > 0. Multiplying each of these series by αi and summing them up over i , we obtain the required inequality
αiφ(xi ) ≥ φ(˜x).
