# CHAPTER 15

# INEQUALITIES

We introduced several important inequalities in Chapter 8 when we presented the *L*^{p} 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 : *G* → **R** 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 : *G* → **R** *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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