414/514 – Convexity

This is the last homework set of the term. It is optional. If you decide to turn it in, it is due Wednesday, December 14 at noon.

1.

We say that a function satisfies Cauchy’s functional equation iff

for all . Show that and that there is a constant such that for all rational . One way to organize the argument is as follows: First show that there is such a if is a natural number, by induction. Then show that the same works if is an integer (positive, negative, or zero), then that it works if is the multiplicative inverse of an integer, and finally for arbitrary rational numbers.

Show that if is continuous, then the same holds for all reals . Show that it suffices to assume that is continuous at 0 to conclude that it is continuous everywhere. What if we assume that is continuous at a point (but could be different from 0)? Can we conclude that is continuous everywhere in this case as well? (On the other hand, using the axiom of choice one can produce examples of discontinuous functions that satisfy Cauchy’s functional equation.)

2.

Suppose that is a function defined on an interval, and that for any and in its domain, and any , we have that

If this is the case, we say that is convex. Draw a diagram explaining the definition. Convexity is a very useful notion in analysis, especially when combined with additional conditions, such as harmonicity.

Show that if is continuous and satisfies the condition above for , then it is convex.

Show that if is convex, then it is continuous, except perhaps at the end points of the interval. In any case, if is convex, continuous, and it is defined at an end point of the interval, show that the limit of as exists (and is finite).

Suppose that is convex and that are points in the interval. Show that

and explain what this says about the graph of .

Show that if is convex and differentiable, then is increasing.

Show that if is convex then whether is differentiable or not, its left and right derivatives exist at every point in the interior of the interval, and are increasing. Here, the left derivative of at is

and similarly for the right derivative (replace with ).

Show that if is twice differentiable on an open interval, then is convex iff for all in the interval.

One of the main reasons why convexity is a useful property is the following theorem of Jensen. Please give a proof: Suppose that is convex and are nonnegative numbers that add up to 1. Then for any in the domain of ,

As an application, prove the inequality between the arithmetic and the geometric means: If are nonnegative numbers, then

(It may be useful to prove first that is a convex function.)

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