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1/3 <= f(1/3) <= 4/7

Source: Nordic Mathematical Contest 2000 #4

October 3, 2017

inequalitiesFunctional inequalitycalculus

Problem Statement

The real-valued function

f

is defined for

0 \le x \le 1, f(0) = 0, f(1) = 1

, and

\frac{1}{2} \le \frac{ f(z) - f(y)}{f(y) - f(x)} \le 2

for all

0 \le x < y < z \le 1

with

z - y = y -x

. Prove that

\frac{1}{7} \le f (\frac{1}{3} ) \le \frac{4}{7}

.

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