Advanced book on Mathematics Olympiad

546 Real Analysis

Ifχ[ 0 ,ai]is the characteristic function of the interval[ 0 ,ai](equal to 1 on the interval and
to 0 outside), then our inequality is, in fact,

∫∞

0

(n

∑

i= 1

xiχ[ 0 ,ai](t)

) 2

dt≥ 0 ,

which is obvious. Equality holds if and only if

∑n
i= 1 xiχ[^0 ,ai]=0 everywhere except at
finitely many points. It is not hard to see that this is equivalent to the condition from the
statement.
(G. Dospinescu)

478.This is just the Cauchy–Schwarz inequality applied to the functionsfandg,g(t)= 1
fort∈[ 0 , 1 ].

479.By Hölder’s inequality,

∫ 3

0

f(x)· 1 dx≤

(∫ 3

0

|f(x)|^3 dx

)^13 (∫ 3

0

1

(^32)
dx

)^23

= 3

(^23)

(∫ 3

0

|f(x)|^3 dx

)^13

.

Raising everything to the third power, we obtain

(∫ 3

0

f(x)dx

) 3 /∫ 3

0

f^3 (x)dx≤ 9.

To see that the maximum 9 can be achieved, choosefto be constant.

480.The argument relies on Figure 69. The left-hand side is the area of the shaded region
(composed of the subgraph offand the subgraph off−^1 ). The productabis the area of
the rectangle[ 0 ,a]×[ 0 ,b], which is contained inside the shaded region. Equality holds
if and only if the two regions coincide, which is the case exactly whenb=f(a).
(Young’s inequality)

y=f(x)

a

b

Figure 69