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China Mathematical Olympiad 1993 problem6

Source: China Mathematical Olympiad 1993 problem6

September 23, 2013

functioninequalitiesinduction

Problem Statement

Let

f: (0,+\infty)\rightarrow (0,+\infty)

be a function satisfying the following condition: for arbitrary positive real numbers

x

and

y

, we have

f(xy)\le f(x)f(y)

. Show that for arbitrary positive real number

x

and natural number

n

, inequality

f(x^n)\le f(x)f(x^2)^{\dfrac{1}{2}}\dots f(x^n)^{\dfrac{1}{n}}

holds.