Functions (continued)

Prove that a function convex on an interval $\Delta$ satisfies Jensen's inequality $f\left(\sum_{1 \leqslant j \leqslant n} \lambda_{j} x_{j}\right) \leqslant \sum_{1 \leqslant j \leqslant n} \lambda_{j} f\left(x_{j}\right)$ for all $x_{1}, \ldots , x_{n} \in \Delta , \lambda_{1}, \ldots , \lambda_{n} \in [0,1], \lambda_{1}+\ldots +\lambda_{n}=1$.

Let the function $f$ be defined on an interval $\Delta$, and let $x_{1}<x_{2}<x_{3}$ be points of $\Delta$. Consider the three chords joining the pairs of points of the graph of $f$ with abscissas $x_{1}$ and $x_{2}, x_{1}$ and $x_{3}, x_{2}$ and $x_{3}$. The slopes of these chords are, respectively, $\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}, \frac{f\left(x_{3}\right)-f\left(x_{1}\right)}{x_{3}-x_{1}}, \frac{f\left(x_{3}\right)-f\left(x_{2}\right)}{x_{3}-x_{2}}$. Prove that for the convexity of the function $f$ it is necessary and sufficient that for any triple of points $x_{1}<x_{2}<x_{3}$ of $\Delta$ the inequalities

hold (the three-chord lemma, see Fig. 8).

Fig. 8

Check that this double inequality is equivalent to

Let the function $f$, defined on an interval $\Delta$, satisfy the inequality $f\left(\frac{x_{1}+x_{2}}{2}\right) \leqslant \frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2}$ for all $x_{1}, x_{2} \in \Delta$. Prove that

Prove that the continuity condition in problem 1.3 b) can be replaced by the following: the function $f$ is bounded above on some nonempty interval $(p, q) \subset \Delta$ (or even on $(p, q) \backslash \mathbb {Q}$).

Prove that for a function $f:(a, b) \rightarrow \mathbb {R}$ the following three statements are equivalent:

1. $f$ is convex;

2. the epigraph of the function $f$

   $$\Gamma ^{+}=\left\{ (x, y) \in \mathbb {R}^{2} \mid x \in (a, b), y \geqslant f(x)\right\}$$

   is a convex set;

3. through every point of the graph of $f$ one can draw a supporting line for the epigraph (i.e., a line such that all points of the epigraph lie not below it).

Prove that if a function defined on an interval is locally convex, then it is convex.

Let $f$ be a function convex on the interval $\left[a, b\right]$. Prove that then

Let the function $f$ be concave on the interval $[0,+\infty ), f(0)=0$. Prove that for all $x, y \geqslant 0$ the inequality

holds.

Prove that if the function $f$ is convex and strictly monotone, then $f^{-1}$ is either convex or concave.

Let the function $f$ be convex, and the function $g$ be convex and increasing. Prove that if the composition $g \circ f$ makes sense, then it is convex.

Let $f \in C((0,+\infty ))$. Prove that then the functions $x \mapsto x f(x), x \mapsto f\left(\frac{1}{x}\right)$ are convex or not simultaneously.

Let the function $f$ be convex on $[a,+\infty )$. Prove that the limit $\lim_{x \rightarrow +\infty } \frac{f(x)}{x}=l \in \overline{\mathbb {R}}$ exists, and moreover $l>-\infty$.

Prove that a convex function $f$, defined on the interval $\langle a, b\rangle$, is continuous on $(a, b)$ and has there finite increasing one-sided derivatives $f_{-}^{\prime }(x), f_{+}^{\prime }(x)$, and $f^{\prime }(x)$ exists everywhere except at no more than a countable set of points. Prove that $f \in \operatorname {Lip}_{1}(\Delta )$ for any closed interval $\Delta \subset (a, b)$.

Prove that a function $f \in C((a, b))$ is convex if and only if for all $x \in (a, b)$ the inequality

holds.

In particular, if $L_{f}(x)=0$ for all $x \in (a, b)$, then $f$ is a linear function; if $f$ is twice differentiable on $(a, b)$, then it is convex if and only if $f^{\prime \prime }(x) \geqslant 0$ on $(a, b)$.

Let $f \in C(\langle a, b\rangle )$. Prove that for the convexity of the function $f$ it is necessary and sufficient that for all $x \in (a, b)$ and such $h>0$ that $x-h, x+h \in \langle a, b\rangle$, the inequality

hold.

Let the function $f$ be convex on $\mathbb {R}$ and $\lim_{x \rightarrow \pm \infty } \frac{f(x)}{x}=0$. Prove that $f$ is constant.

The following problem often allows one, when proving various statements about convex functions, to additionally assume their smoothness (in particular, some of the preceding problems can be solved this way).

Prove that every continuous function convex on $\left[a, b\right]$ is the limit of a uniformly convergent decreasing sequence of:

Prove that if the function $f:[0,+\infty ) \rightarrow \mathbb {R}$ is differentiable, concave, and nonnegative, then $x f^{\prime }(x) \leqslant f(x)$ for all $x \geqslant 0$.

Let the function $f$ be convex and increasing on the interval $[0,+\infty ), f(0)=0$. Prove that for any numbers $a_{0} \geqslant a_{1} \geqslant \ldots \geqslant a_{n} \geqslant 0$ the inequality

holds.

A sequence $\left\{ a_{n}\right\} \subset \mathbb {R}$ is called convex if $\Delta^{2} a_{n}=a_{n+2}-2 a_{n+1}+a_{n} \geqslant 0$, i.e., $a_{n+1} \leqslant \frac{1}{2}\left(a_{n}+a_{n+2}\right)$ for any $n \in \mathbb {N}$. Prove that if the sequence $\left\{ a_{n}\right\}$ is convex, then

Let $\left\{ a_{n}\right\}_{n \geqslant 0}$ be a bounded convex sequence (see problem 1.20). Prove that

Let $L>0, f \in C(\left[0, L\right]), f \geqslant 0$. Set $M=\int_{0}^{L} f(x) d x$,

Prove that if the function $f$ is concave, then

We keep the notation of the previous problem, but we shall consider not concave but convex functions.

Let the function $f$ be nondecreasing on $\left[0, \frac{1}{2}\right]$ and $f(1-x)=f(x)$ on $\left[0,1\right]$. Prove that for any function $\varphi$ convex on $\left[0,1\right]$ the inequality

holds.

Let the functions $f, g$ be nonnegative and convex on the interval $\left[a, b\right]$, with one of the endpoints of the interval being a root of both functions. Prove the inequality

Let the functions $f$ and $g$ be nonnegative and concave on the interval $\left[a, b\right]$. Prove the inequality

Let $0<a<b, K$ be the set of all nonnegative nonincreasing functions $f$ on the interval $\left[a, b\right]$ such that

Prove that for any functions $f$ and $g$ from $K$ the inequality

holds.

For which functions is equality attained?

Let the function $f$ be integrable on the interval $\left[a, b\right]$. Prove that the nonnegativity of the integral $\int_{a}^{b} f(x) \varphi (x) d x$ for any function $\varphi$ convex on $\left[a, b\right]$ is equivalent to the function $f$ satisfying three conditions:

1. $\int_{a}^{b} f(x) d x=0$;

2. $\int_{a}^{b} x f(x) d x=0$;

3. $\int_{a}^{c}(c-x) f(x) d x \geqslant 0$ for any $c \in (a, b)$.

Let the function $f$ decrease to zero on $(0,+\infty )$ and $a>0$. Prove that $\int_{0}^{\infty } f(x) \sin a x d x \geqslant 0$, and if $f$ is convex, then $\int_{0}^{\infty } f(x) \cos a x d x \geqslant 0$. In both cases equality is possible only for $f \equiv 0$.

Let $f \in C((0,+\infty )), f(1)=1$. Prove that the system of inequalities

where $x$ and $y$ are arbitrary positive numbers, is satisfied only by the power function $f(x)=x^{p}$ with exponent $p \in [0,1]$, and if both inequality signs are replaced by $\leqslant$, then only by the power function with exponent $p \notin (0,1)$. Verify that this is not the case if the sign is replaced only in the second inequality.

Let $a_{j}, b_{j} \geqslant 0, j=1, \ldots , n ; p, q>1, \frac{1}{p}+\frac{1}{q}=1$. Prove Hölder's inequality

Let $x_{j}>0, j=1, \ldots , n, p \neq 0$. Set

Prove that

Let the piecewise continuous functions $f$ and $p$ on $\left[a, b\right]$ satisfy the conditions $f(\left[a, b\right]) \subset [m, M], p \geqslant 0, \int_{a}^{b} p(x) d x=1$, and let the function $\varphi$ be convex on $\left[m, M\right]$. Prove that

Let $A_{0}, A_{1}, \ldots , A_{n}$ be the vertices of a convex polygon inscribed in a circle, with the vertices $A_{0}$ and $A_{n}$ fixed. For a given $n$, how should one choose the points $A_{1}, \ldots , A_{n-1}$ so that

A light source placed at the point $(0, b), b>0$, illuminates a region that is the subgraph of a nonnegative convex function $f \in C^{1}([0,+\infty )), f(0)>b$. Light rays are reflected from the graph of $f$ and from the abscissa axis according to the known law. Will the entire region be illuminated if $\lim_{x \rightarrow +\infty } f(x)=0$?

Let the function $f: \mathbb {R} \rightarrow (-\infty ,+\infty ]$ be convex, let $D(f)$ be the interval $\left\{ x \in \mathbb {R} \mid f(x)<+\infty \right\}$, and let $\Delta$ be the interval $\left(\inf f^{\prime }(t), \sup f^{\prime }(t)\right)$, where the infimum and supremum are taken over the set of points $t$ at which $f^{\prime }(t)$ exists.

The Legendre transform of the function $f$ is the function

The quantity $f^{*}(x)$ shows how far one must shift vertically the line through the origin with slope $x$ so that it becomes a supporting line to the epigraph of the function $f$ (Fig. 9).

Fig. 9

Prove that

Prove that if the function $\varphi$ is strictly increasing and continuous on $[0,+\infty )$, $\varphi (0)=0$, $\psi =\varphi^{-1}$, then for $a \in [0,+\infty )$ and $b \in [0, \sup \varphi )$ the inequality

holds.

Find the Legendre transform and write Young's inequality (see problem 1.37 d)) for the following functions:

Consider the collection $W$ of pairs $(f, g)$ of functions convex on $\mathbb {R}$ satisfying the condition $x y \leqslant f(x)+g(y)$ for all $x, y \in \mathbb {R}$. Call a pair $\left(f_{0}, g_{0}\right) \in W$ extremal if the relations $f \leqslant f_{0}, g \leqslant g_{0},(f, g) \in W$ imply $f=f_{0}, g=g_{0}$. Prove that the pair ($f_{0}, g_{0}$) is extremal if and only if $f_{0}^{*}=g_{0}$, $g_{0}^{*}=f_{0}$.

Give examples of functions for which $f^{*}(x)=f(-x) (x \in \mathbb {R})$. Prove that the only function coinciding with its own Legendre transform is the function $f(t)=\frac{t^{2}}{2}$.

Prove that if the function $f$ is $m$ times differentiable on the interval $(a, b)$ and $x, x+m h \in (a, b)$, then

where $0<\theta <m$ (for the definition of $\Delta_{h}^{m} f(x)$ see problem III.3.1).

Let the function $f$ be $n$ times differentiable on $\left[0,1\right]$ and $f^{(k)}(0)= =f^{(k)}(1)=0$ for $k=1, \ldots , n-1$. Prove that for some $x \in [0,1]$ the inequality

holds.

Let the function $f$ be continuous on $\mathbb {R}$, and suppose there exist real numbers $a_{k}, b_{k}, c_{k}(1 \leqslant k \leqslant n)$ such that

Prove that if $c_{k} \neq 0$ for all $k$, then $f$ is a polynomial.

Let $f(x)=P_{1}(x) e^{Q_{1}(x)}+\ldots +P_{n}(x) e^{Q_{n}(x)}$, where $P_{k}, Q_{k} (k=1,2, \ldots , n)$ are real polynomials, $x \in \mathbb {R}$. Prove that if $f \not\equiv 0$, then $f$ has a finite number of zeros.

Let $f \in C^{\infty }([0, \infty ))$, $f(0)=\lim_{x \rightarrow +\infty } f(x)$. Prove that for each $n \in \mathbb {N}$ the function $f^{(n)}$ vanishes.

Let $f \in C^{2}((0,1))$ and $f(x)=o(1)$, $f^{\prime \prime }(x)=O\left(x^{-2}\right)$ as $x \rightarrow +0$. Prove that $f^{\prime }(x)=o\left(x^{-1}\right)$ as $x \rightarrow +0$.

Let $\Delta$ be an interval, $f \in C^{2}(\Delta )$, $M_{k}=\sup_{\Delta }\left|f^{(k)}\right|, k=0,1,2$.

Let $f \in C^{2}([0,+\infty ))$ and $\int_{0}^{+\infty }\left|f(x)\right| d x<+\infty$. Prove the inequalities:

Find $\min \int_{0}^{1}\left|f^{\prime \prime }(x)\right|^{2} d x$ subject to the condition that $f \in C^{2}(\left[0,1\right])$, $f(0)=f(1)=0, f^{\prime }(0)=a$.

Let $f \in C^{2}(\left[0,1\right]), f(0)=f(1)=0$ and $\left|f^{\prime \prime }(x)\right| \leqslant 1$ for all $x \in [0,1]$. Prove the inequalities:

Let $f \in C^{\infty }(\left[a, b\right]), f \not\equiv 0$. Prove that if the function $f$ vanishes on some nonempty interval, then

Does there exist a point $x$ such that $\sup_{n}\left|f^{(n)}(x)\right|=+\infty$?

Let $f \in C^{\infty }(\left[a, b\right])$. Prove that if the sequence of derivatives $\left\{ f^{(n)}(x)\right\}_{n \geqslant 0}$ is bounded at every point $x \in [a, b]$, then it is uniformly bounded: $\left|f^{(n)}(x)\right| \leqslant C$, where $C$ does not depend on $n$ or $x$.

Let $f \in C^{\infty }(\left[a, b\right])$ and suppose that everywhere on $\left[a, b\right]$ the finite limit $\varphi (x)=\lim f^{(n)}(x)$ exists. Prove that the function $\varphi$ has the form $\varphi (x)=$ const $\cdot e^{x}$.