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ineq function

Source: Vietnam TST 2000

April 3, 2007

functionalgebra unsolvedalgebra

Problem Statement

Let

a > 1

and

r > 1

be real numbers. (a) Prove that if

f : \mathbb{R}^{+}\to\mathbb{ R}^{+}

is a function satisfying the conditions (i)

f (x)^{2}\leq ax^{r}f (\frac{x}{a})

for all

x > 0

, (ii)

f (x) < 2^{2000}

for all

x < \frac{1}{2^{2000}}

, then

f (x) \leq x^{r}a^{1-r}

for all

x > 0

. (b) Construct a function

f : \mathbb{R}^{+}\to\mathbb{ R}^{+}

satisfying condition (i) such that for all

x > 0, f (x) > x^{r}a^{1-r}

.

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