# INEQUALITIES

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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