Homework Assignment 4, MATH 515, Spring 09
Problem 8) (4 pts) A subset Dof a real vector space Eis convex if for all
x, y ∈Dand 0 < t < 1 it follows tx + (1 −t)y∈D. Show the following: If
x1, . . . , xn∈Dthen also t1x1+. . . +tnxn∈Dfor all t1, . . . , tn>0 such that
Pn
i=1 ti= 1. Conversely, for given x1, . . . , xn∈Ethe set of all combinations as
above is a convex set.
Problem 9) (12 pts) (a) Let D⊂Rnbe convex. A function f:D→Ris
called convex if f(tx+(1−t)y)≤tf(x)+(1−t)f(y) for all x, y ∈Dand 0 <t<1.
The function fis called strictly convex if strict inequality holds whenever x6=y
and 0 <t<1. The function fis concave respectively strictly concave if −f
is convex respective strictly convex. Show the following: If f: (a,b)→Ris
differentiable then fis convex ⇐⇒ f0is increasing, and fis concave ⇐⇒ f0
is decreasing. In particular, if fis twice differentiable and f00 ≥0 then fis
convex, while if f00 ≤0 then fis concave. Also f00 >0 then fis strictly convex
and if f00 <0 then fis strictly concave.
(b) Prove Jensen’s inequality: For Da convex subset of a vector space Eand
f:D→Ra concave function show that
n
X
i=1
tif(xi)≤f n
X
i=1
tixi!
whenever x1,...xn∈D,t1, . . . , tn∈(0,1) and Pn
i=1 ti= 1. Furthermore, if
fis strictly concave then equality holds if and only if x1=.. . =xn. Hint:
induction on n.
(c) Apply (b) to f(x) = log(x) to show that if a1,...an≥0 and p1, . . . , pn>0
are real numbers with Pn
i=1 pi= 1 then
n
Y
i=1
api
i≤
n
X
i=1
piai
with equality if and only if a1=. . . =an.
(d) Prove that for a, b ≥0 and p, q > 1 with 1
p+1
q= 1 it follows ab ≤ap
p+bq
q
with equality if and only if ap=bq.
Problem 10 (8 pts) Prove H¨older’s inequality: Suppose p, q > 1 and 1
p+1
q= 1.
Then for complex numbers a1, . . . , an, b1, . . . , bnwe have
n
X
k=1
akbk
≤ n
X
k=1
|ak|p!1/p n
X
k=1
|bk|q!1/q
with equality if and only if all akare 0 or |bk|q=t|ak|pand akbk=eiθ|akbk|for
all kand some tand θ. Hints: Reduce to the case Pn
k=1 |ak|p=Pn
k=1 |bk|q= 1.