Math Problems Vault

Problem 1.8.1

Let $f : [0,1] \rightarrow \mathbb {R}$ be continuous with $f(0) = 0$ . Show there is a continuous concave function $g : [0,1] \rightarrow \mathbb {R}$ such that $g(0) = 0$ and $g(x) \geq f(x)$ for all $x \in [0,1]$ .

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Remark.

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A function $g : I \rightarrow \mathbb {R}$ is concave if

g(tx+(1-t)y) \geq tg(x)+(1-t)g(y)

for all $x$ and $y$ in $I$ and $0 \leq t \leq 1$ .

Problem 1.8.2

Let $f : I \rightarrow \mathbb {R}$ (where $I$ is an interval of $\mathbb {R}$ ) be such that $f(x) > 0$ , $x \in I$ . Suppose that $e^{cx}f(x)$ is convex in $I$ for every real number $c$ . Show that $\log f(x)$ is convex in $I$ .

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Remark.

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A function $g : I \rightarrow \mathbb {R}$ is convex if

g(tx+(1-t)y) \leq tg(x)+(1-t)g(y)

for all $x$ and $y$ in $I$ and $0 \leq t \leq 1$ .

Problem 1.8.3

Let $f$ be a real valued continuous function on $\mathbb {R}$ satisfying the mean value inequality below:

f(x) \leq \frac{1}{2h} \int _{x-h}^{x+h} f(y)\, dy, \quad x \in \mathbb {R}, \quad h > 0.

Prove:

The maximum of $f$ on any closed interval is assumed at one of the endpoints.
$f$ is convex.

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