3.4 Equations with Functions as Unknowns 187Example.Letx 1 ,x 2 ,...,xnbe arbitrary real numbers. Prove the inequality
x 1
1 +x^21+
x 2
1 +x^21 +x 22+···+
xn
1 +x 12 +···+xn^2<
√
n.Solution.We introduce the function
fn(x 1 ,x 2 ,...,xn)=x 1
1 +x 12+
x 2
1 +x 12 +x 22+···+
xn
1 +x 12 +···+x^2 n.
If we setr=
√
1 +x 12 , thenfn(x 1 ,x 2 ,...,xn)=
x 1
r^2+
x 2
r^2 +x 22+···+
xn
r^2 +x 22 +···+x^2 n=x 1
r^2+
1
r( x
2
r
1 +(xr^2 )^2+···+
xn
r
1 +(xr^2 )^2 +···+(xn
r) 2
)
.
We obtain the functional equation
fn(x 1 ,x 2 ,...,xn)=x 1
1 +x 12+
1
√
1 +x 12fn− 1(x 2
r,
x 3
r,...,
xn
r)
.
WritingMn=supfn(x 1 ,x 2 ,...,xn), we observe that the functional equation gives rise
to the recurrence relation
Mn=sup
x 1⎛
⎝ x^1
1 +x 12+
Mn− 1
√
1 +x 12⎞
⎠.
We will now prove by induction thatMn<
√
n. Forn=1, this follows from 1 +x^1 x 2
1≤
1
2 <1. Assume that the property is true forkand let us prove it fork+1. From the
induction hypothesis, we obtain
Mk<sup
x 1⎛
⎝ x^1
1 +x 12+
√
k
√
1 +x 12⎞
⎠.
We need to show that the right-hand side of the inequality is less than or equal to
√
k+1.
Rewrite the desired inequality as
x
√
1 +x^2+
√
k≤√
k+kx^2 + 1 +x^2.