Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 157

475.Letf:[ 0 , 1 ]→Rbe a continuous function such that

∫ 1

0

f(x)dx=

∫ 1

0

xf (x)dx= 1.

Prove that

∫ 1

0

f^2 (x)dx≥ 4.

476.For each continuous functionf:[ 0 , 1 ]→R, we defineI(f)=

∫ 1

0 x

(^2) f(x)dxand
J(f)=

∫ 1

0 x(f (x))

(^2) dx. Find the maximum value ofI(f)−J(f)over all such
functionsf.
477.Leta 1 ,a 2 ,...,anbe positive real numbers and letx 1 ,x 2 ,...,xnbe real numbers
such thata 1 x 1 +a 2 x 2 +···+anxn=0. Prove that
∑
i,j
xixj|ai−aj|≤ 0.
Moreover, prove that equality holds if and only if there exists a partition of the set
{ 1 , 2 ,...,n}into the disjoint setsA 1 ,A 2 ,...,Aksuch that ifiandjare in the
same set, thenai=ajand also

∑

j∈Aixj=0 fori=^1 ,^2 ,...,k.
We now list some fundamental inequalities. We will be imprecise as to the classes
of functions to which they apply, because we want to avoid the subtleties of Lebesgue’s
theory of integration. The novice mathematician should think of piecewise continuous,
real-valued functions on some domainDthat is an interval of the real axis or some region
inRn.

The Cauchy–Schwarz inequality.Letfandgbe square integrable functions. Then

(∫

D

f (x)g(x)dx

) 2

≤

(∫

D

f^2 (x)dx

)(∫

D

g^2 (x)dx

)

.

Minkowski’s inequality.Ifp> 1 , then

(∫

D

|f(x)+g(x)|pdx

)^1 p

≤

(∫

D

|f(x)|pdx

)p^1

+

(∫

D

|g(x)|pdx

)^1 p

.

Hölder’s inequality.Ifp, q > 1 such thatp^1 +^1 q= 1 , then

∫

D

|f (x)g(x)|dx≤

(∫

D

|f(x)|pdx

)p 1 (∫

D

|g(x)|qdx

) (^1) q
.