18.099b Problem Set 6 (Corrected version) Due: Thursday, April 15th.

This assignment is to write a short paper on convex functions. It should be typeset in Tex. Arrange and express the material discussed below in whatever way you think best. The paper should flow smoothly and be well motivated.

A real valued function f on an open (possibly unbounded) interval I ⊆ ℝ is convex if for all x, y ∈ I

f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y)

for all 0 < λ < 1. Explain the geometric meaning of this definition; namely that the part of the graph of the function between any two points x, y ∈ I lies below the chord joining (x, f(x)) and (y, f(y)).

Make precise and prove the statement that an increasing convex function applied to a convex function is again convex.

Show that if f is continuous and f(\frac{x+y}{2} ) ≤ \frac{f(x)+f(y)}{2} for all x, y ∈ I. then f is convex. Hint: Use the fact that if g and h are continuous functions with g(a) > h(a) for some a, then g > h on (a − δ, a + δ) for some δ > 0.

Prove that f is convex on I if and only if for every x < t < y in I

(1)  (y − x)f(t) ≤ (y − t)f(x) + (t − x)f(y).

In particular if f is convex and a ∈ I, then the function \frac{f(t) − f(a)}{t − a} is increasing on

I \{a}. Conclude that every convex function is continuous and has both right-hand and left-hand derivatives at every point in I.

Suppose f is convex and differentiable on I. Show that for every a ∈ I, the graph of f lies above the tangent line to the graph at (a, f(a)).

Suppose f is differentiable on I. Show that f is convex if and only if f′ is increasing. Hint: Use the mean value theorem and the inequality (1).

Give (interesting) examples of convex elementary functions.

Prove Jensen’s inequality: If f is convex on I, x_{1}, …, x_{n} ∈ I, and λ_{1}, …, λ_{n} are

non-negative real numbers such that ∑_{i=1}^{n} λ_{i} = 1, then

f(∑_{i=1}^{n} λ_{i}x_{i}) ≤ ∑_{i=1}^{n} λ_{i}f(x_{i}).

(Notice that in order for this to even make sene, ∑_{i=1}^{n} λ_{i}x_{i} must also be in I. Why

is this the case?).

As an application of Jensen’s inequality, deduce the “inequality of the arithmetic and geometric means”: if x_{1}, …, x_{n} are positive real numbers, then

(x_{1}x_{2} ⋯ x_{n})^{\frac{1}{n}} ≤ \frac{1}{n} ∑_{i=1}^{n} x_{i}.