We introduced several important inequalities in Chapter 8 when we presented the Lp spaces. In this chapter, we shall introduce Jensen’s inequality for convex functions. Jensen’s inequality can be used to prove many other inequalities.

15.1 Basic Concepts and Facts

Definition 15.1 (Convex Functions). Let G be an open subinterval of R. A function φ : GR is considered convex on G if its graph lies below any of its chords; that is, for any x, y G and 0 ≤ p ≤ 1, we have


where q = 1 − p.

Theorem 15.1 (Mean Value Theorem). Let f(x) be a continuous function on the closed interval [a, b]. Suppose that f(x) is differentiable on the open interval (a, b). Then there exists a point c (a, b) such that


Theorem 15.2 (Jensen’s Inequality). Let (Ω, , P) be a probability space and φ : GR be a convex function on G, where G is an open subinterval of R. Suppose that X is a random variable on Ω such that E(|X|) < ∞, P(X G) = 1, and E(|φ(X)|) < ∞. Then

15.2 Problems

15.1 (Chebyshev’s ...

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