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{\bf Freshman Seminar 24i: Mathematical Problem Solving}
\\
Using AM--GM and other applications of Jensen's inequality
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Some simple applications:

1) Of all rectangular boxes of volume~$V\!$, the cube has the smallest
surface area.

2) An ``open box'' is a rectangular box missing one of its six sides
(so that we can put things in and out of the box).  Given~$V\!$,
what is the open box of volume~$V$\/ that has minimal surface area?
Upon further reflection, can you derive this directly from~(1)?

3) Of all triangles of area~$A$, the equilateral triangle has
the smallest perimeter.

Some proofs of the inequality and generalizations:

4) Say a real-valued function $f$\/ on some (possibly infinite)
interval~$I$\/ is ``midpoint-convex'' upwards if
$$
f\left(\frac12(x+y)\right) \leq \frac12\Bigl(f(x)+f(y)\Bigr)
$$
for all $x,y$ in~$I$\/; we say $f$\/ is {\em strictly} midpoint-convex
upwards if equality holds only for $x=y$.  Prove that if~$f$\/ is
% midpoint-convex upwards then Jensens' inequality
midpoint-convex upwards then Jensen's inequality
$$
f\left(\frac1n(x_1+x_2+\cdots+x_n)\right) \leq
\frac1n\Bigl( f(x_1) + f(x_2) + \cdots + f(x_n) \Bigr)
$$
holds for all $n$ and for all $x_1,\ldots,x_n$ in~$I$\/;
and that if $f$\/ is strictly midpoint-convex then equality holds
if and only if all the $x_i$ are equal.
[Hint: the inequality is trivial for $n=1$ and the definition of
midpoint-convexity for $n=2$.  Prove it first for $n=4,8,16,\ldots$;
to get the general case, try first to derive the $n=3$ case from $n=4$.
Thanks to Zach for reminding me of the AM--GM case of this trick.]

5) Show directly (without calculus etc.)\ that the functions
$x^2$ and $e^x$ are midpoint-convex upwards on all of~$\bf R$,
that $1/x$ is midpoint-convex upwards on $x>0$,
and that $\sin x$ is midpoint-convex downwards on $0 \leq x \leq \pi$.
(Hint: it can be surprisingly useful that $a \geq b$\/ if and only if
$a-b \geq 0$.)  Jensen's inequality for these functions then follows from
the previous problem. 

6) Suppose $w_1,w_2,\ldots,w_n$ are nonnegative numbers with
$w_1 + w_2 + \cdots + w_n = 1$.  The {\em weighted average} of 
$z_1, z_2, \ldots, z_n$ with weights $w_1, w_2, \ldots ,w_n$ is
$w_1 z_1 + w_2 z_2 + \cdots + w_n z_n$.  (For instance, the usual
weighted average is recovered by setting each $w_i$ equal $1/n$,
and the barycentric coordinates of a point~$P$\/ in triangle $ABC$
are the weights for which $P$\/ is the weighted average of $A,B,C$\/
--- that's one reason I didn't restrict $z_1,\ldots,z_n$ to
real numbers).  {\em Jensen's inequality for weighted averages}\/
states that
$$
% f(w_1 z_1 + w_2 z_2 + \cdots + w_n z_n) \leq
f(w_1 x_1 + w_2 x_2 + \cdots + w_n x_n) \leq
w_1 f(x_1) + w_2 f(x_2) + \cdots + w_n f(x_n)
$$
if $f$\/ is convex upwards, and likewise
$$
f(w_1 x_1 + w_2 x_2 + \cdots + w_n x_n) \geq
w_1 f(x_1) + w_2 f(x_2) + \cdots + w_n f(x_n)
$$
if $f$\/ is convex downwards.  (Again the inequality compares
the function of the average with the average of the function.)
Prove this inequality: (i) in the same graphical way that we did for the
unweighted version; (ii) as a consequence of Jensen
(first do the case of rational $w_i$, then use continuity).
If $f$\/ is strictly convex, when does equality hold?

7) Show that weighted Jensen still reduces to Cauchy--Schwarz
for $f(x) = x^2$ or $f(x) = 1/x$.

8) Suppose $P$\/ is a polynomial with positive coefficients.
Prove that $P(x)P(y) \geq \left(P(\sqrt{xy}\,)\right)^2$
for all $x,y>0$.  When does equality hold?
What are the functions whose convexity you can deduce from
that inequality?

9) Suppose $f(x)$ is convex downwards on $a \leq x \leq b$
and we fix some $n\geq 1$ and $s$ between $na$ and~$nb$.
Consider the sum $f(x_1) + \cdots + f(x_n)$ subject to
$a \leq x_i \leq b$ and $x_1 + \cdots + x_n = s$.
Jensen tells us that this is maximized when each $x_i$ equals $s/n$.
What choice of $x_1,\ldots,x_n$ {\em minimizes}\/ the sum?
For instance, how small can
$\sin \alpha + \sin \beta + \sin \gamma + \sin \delta + \sin \epsilon$
get if $0 \leq \alpha,\beta,\gamma,\delta,\epsilon \leq \pi/2$ and
$\alpha + \beta + \gamma + \delta + \epsilon = 5$?

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