Question 49
Let $f$ be a function defined on the positive reals with the following properties:
(1) $f(1) = 1$,
(2) $f(x + 1) = x f(x)$, and
(3) $f(x) = 10^{g(x)}$,
where $g(x)$ is a function defined on the reals satisfying
$$
g(ty + (1-t)z) \leq t g(y) + (1-t) g(z)
$$
for all $y$ and $z$ and for $0 \leq t \leq 1$.
a. Prove that $t [g(n) - g(n-1)] \leq g(n+t) - g(n) \leq t [g(n+1) - g(n)]$ where $n$ is an integer and $0 \leq t \leq 1$.
b. Prove that $\frac{4}{3} \leq f(\frac{1}{2}) \leq \frac{4}{3} \sqrt{2}$.
(1) $f(1) = 1$,
(2) $f(x + 1) = x f(x)$, and
(3) $f(x) = 10^{g(x)}$,
where $g(x)$ is a function defined on the reals satisfying
$$
g(ty + (1-t)z) \leq t g(y) + (1-t) g(z)
$$
for all $y$ and $z$ and for $0 \leq t \leq 1$.
a. Prove that $t [g(n) - g(n-1)] \leq g(n+t) - g(n) \leq t [g(n+1) - g(n)]$ where $n$ is an integer and $0 \leq t \leq 1$.
b. Prove that $\frac{4}{3} \leq f(\frac{1}{2}) \leq \frac{4}{3} \sqrt{2}$.