Inequalities

Prove that for any finite sequence $\left\{ a_{k}\right\}_{k=1}^{n}$ of real numbers

Let $\left\{ a_{k}\right\}_{k=1}^{n}$ and $\left\{ b_{k}\right\}_{k=1}^{n}$ be finite nonnegative sequences, with the sequence $\left\{ a_{k}\right\}$ nonincreasing; $B=\max_{1 \leqslant k \leqslant n} b_{k}$. Prove that if the indices $m, M$ do not exceed $n$ and satisfy the inequality $m B \leqslant \sum_{1 \leqslant k \leqslant n} b_{k} \leqslant M B$, then

Let $\left\{ a_{k}\right\}_{k=1}^{n}$ be a finite sequence of positive numbers and $M=\max_{1 \leqslant k \leqslant n} a_{k}, m=\min_{1 \leqslant k \leqslant n} a_{k}$. Prove that

Let $\left\{ q_{k}\right\}_{k=1}^{n}$ be a finite sequence of numbers, with $q_{k} \geqslant 1$ for all $k$. Prove the inequality

Let $\left\{ a_{k}\right\}_{k=1}^{n}$ be a finite sequence of nonnegative numbers, with $S=a_{1}+a_{2}+\ldots +a_{n}<1$. Prove the inequalities

the right-hand inequalities are strict if $S>0$; the left-hand inequalities are strict if at least two of the numbers $\left\{ a_{k}\right\}_{k=1}^{n}$ are positive.

Let $\left\{ a_{k}\right\}_{k=1}^{n}$ be a finite sequence of real numbers, with $a_{k}>-1$ for $k=1, \ldots , n$. Prove that

Let $\left\{ a_{k}\right\}_{k=1}^{n}$ and $\left\{ b_{k}\right\}_{k=1}^{n}$ be finite sequences of nonnegative numbers. Prove that

Let $\left\{ z_{k}\right\}_{k=1}^{n}$ be a finite sequence of complex numbers, $S_{n}=\left|z_{1}\right|+\ldots +\left|z_{n}\right|$ and $P_{n}=\left(1+z_{1}\right) \ldots \left(1+z_{n}\right)$. Prove that $\left|P_{n}\right| \leqslant e^{S_{n}}$ and $\left|1-P_{n}\right| \leqslant S_{n} e^{S_{n}}$.

Let $t, p>0$. Prove that for all $n \in \mathbb {N}$ the inequalities

hold.

For which $p \leqslant 1$ are these inequalities sharp in order as $n$ increases without bound?

Prove that

$\left\{ p_{n}\right\}$ is the sequence of prime numbers, numbered in increasing order ($p_{1}=2$). Let $s>0$. Prove that

$\left\{ p_{n}\right\}$ is the sequence of prime numbers, numbered in increasing order ($p_{1}=2$). Prove that for $s>1$ the following inequalities hold:

$\left\{ p_{n}\right\}$ is the sequence of prime numbers, numbered in increasing order ($p_{1}=2$). Let $\pi (n)$ be the number of primes not exceeding $n$. Verify that

Deduce from this that for all $n \in \mathbb {N}$ the following inequalities hold:

$\left\{ p_{n}\right\}$ is the sequence of prime numbers, numbered in increasing order ($p_{1}=2$). Let $Q_{n}$ be the least common multiple of the numbers $n+1, n+2, \ldots$, $2 n+1$. Verify that

$\left\{ p_{n}\right\}$ is the sequence of prime numbers, numbered in increasing order ($p_{1}=2$). Prove that

For finite sequences of real numbers $\left\{ a_{k}\right\}_{k=1}^{n}$ and $\left\{ b_{k}\right\}_{k=1}^{n}$, denote by $\left\{ \hat{a}_{k}\right\}_{k=1}^{n}$ and $\left\{ \hat{b}_{k}\right\}_{k=1}^{n}$ (respectively $\left\{ \stackrel{V}{a}_{k}\right\}_{k=1}^{n}$ and $\left\{ \stackrel{V}{b_{k}}\right\}_{k=1}^{n}$) the nondecreasing (respectively nonincreasing) rearrangements of these sequences. Prove the inequalities

Let $\left\{ b_{k}\right\}_{k=1}^{n}$ be a finite sequence of real numbers. Prove that for any finite nonnegative nonincreasing sequence $\left\{ a_{k}\right\}_{k=1}^{n}$ the following inequalities hold:

Let $\left\{ a_{k}\right\}_{k=1}^{n}$ be a finite sequence of real numbers. Prove that

where $\Delta =\max_{1 \leqslant k<n}\left|a_{k}-a_{k+1}\right|, \delta =\min_{1 \leqslant j<k \leqslant n}\left|a_{k}-a_{j}\right|$. In which cases do these inequalities become equalities?

Let $\left\{ a_{k}\right\}_{k=1}^{n}$ be a nondecreasing convex (or nonincreasing concave) sequence of real numbers. Prove the inequalities

If the sequence $\left\{ a_{k}\right\}_{k=1}^{n}$ is nonincreasing and convex (or nondecreasing and concave), then the inequality signs should be reversed.

The Chebyshev polynomial $T_{n}$ is the polynomial defined on the interval $\left[-1,1\right]$ by the equality

(the graphs of $T_{n}$ for $n=1,2,3$ are shown in Fig. 4, and for $n=8$ in Fig. 5).

Fig. 4

Fig. 5

Prove that

This problem uses the Chebyshev polynomials $T_{n}(x)=\cos (n \arccos x)$ (see problem 2.20). Prove the extremal property of Chebyshev polynomials: if $P$ is an algebraic polynomial of degree $n$ (with real or complex coefficients) and $\left|P(x)\right| \leqslant 1$ for all $x \in [-1,1]$, then $\left|P^{(j)}(x)\right| \leqslant \left|T_{n}^{(j)}(x)\right|$ for all $j=0,1, \ldots , n$ and $x \in \mathbb {R},\left|x\right|>1$. The inequality is strict if $P \neq c T_{n}$, where $c \in \mathbb {C},\left|c\right|=1$.

In particular, for $j=0$ we obtain $\left|P(x)\right| \leqslant \left|T_{n}(x)\right|$ outside the interval $\left[-1,1\right]$, and the inequality for $j=n$ implies that the leading coefficient of the polynomial $P$ is, in absolute value, at most $2^{n-1}$, with equality possible only for $P=c T_{n}$, where $c \in \mathbb {C},\left|c\right|=1$.

This problem uses the Chebyshev polynomials $T_{n}(x)=\cos (n \arccos x)$ (see problem 2.20). Let $P(\varphi )=a_{0}+\sum_{k=1}^{n}\left(a_{k} \cos k \varphi +b_{k} \sin k \varphi \right)$ be a trigonometric polynomial of order $n$ with real coefficients $a_{0}, a_{1}, \ldots , a_{n}$ and $b_{1}, \ldots , b_{n}$. Set $M=\max_{\varphi \in \mathbb {R}}\left|P(\varphi )\right|$. Prove Bernstein's inequality:

Deduce from this Bernstein's inequality for algebraic polynomials:

where $P$ is an algebraic polynomial of degree $n$ with real coefficients, $M=\max_{\left|x\right| \leqslant 1}\left|P(x)\right|$.

In which cases do these inequalities turn into equalities at at least one point?

Let $n \in \mathbb {N}$. Prove the inequalities

Let $n \in \mathbb {N}$ and $\varphi_{n}(x)=e^{-\frac{x^{2}}{2 n}}-e^{x}\left(1-\frac{x}{n}\right)^{n}$. Prove that the function $\varphi_{n}$ is nonnegative on the interval $\left[0, n\right]$ and

The next problem considers relations that supplement the classical inequalities

Prove the inequalities

Prove that on the interval $\left[-1,1\right]$ the inequalities hold

Prove the inequalities

Prove the inequalities

Prove that for any $x \in \mathbb {R}$ and $n \in \mathbb {N}$ the inequalities hold

Prove that $\left|\left(\frac{\sin x}{x}\right)^{\prime }\right| \leqslant \frac{1}{k \pi }$ for $x \in [k \pi , k \pi +\pi ](k \in \mathbb {N})$.

Prove that for any $x \in [0,1]$ the following inequalities hold

Let $f, g \in C(\left[0,1\right])$, where the function $f$ is nondecreasing and $0 \leqslant f, g \leqslant 1$ on $\left[0,1\right]$. Prove the inequality

The integral inequalities considered here have obvious summation analogues; their proofs can be obtained either by passing to the limit in the summation inequality, or by modifying the proofs used in the discrete case. Let the function $f$ be integrable on the interval $\left[a, b\right]$ and $0<m=\inf f, M=\sup f$. Then the following inequalities hold:

The integral inequalities considered here have obvious summation analogues; their proofs can be obtained either by passing to the limit in the summation inequality, or by modifying the proofs used in the discrete case. By means of passage to the limit these inequalities carry over to the case of improper integrals over the interval $[a,+\infty )$. Let the functions $f$ and $g$ be integrable on the interval $\left[a, b\right]$, with $f \downarrow , f \geqslant 0$. For $x \in [a, b]$ set $G(x)=\int_{a}^{x} g(t) d t$. Then the following inequalities hold:

The integral inequality considered here has an obvious summation analogue; its proof can be obtained either by passing to the limit in the summation inequality, or by modifying the proof used in the discrete case. Let the functions $f$ and $g$ be integrable on the interval $\left[a, b\right]$, with $f \downarrow$ and $0 \leqslant g \leqslant 1$. Set $c=\int_{a}^{b} g(x) d x$. Then

The integral inequality considered here has an obvious summation analogue; its proof can be obtained either by passing to the limit in the summation inequality, or by modifying the proof used in the discrete case. Let $f \in C^{1}(\left[a, b\right]), \delta =\min_{\left[a, b\right]}\left|f^{\prime }\right|, \Delta =\max_{\left[a, b\right]}\left|f^{\prime }\right|$. Then

Equality holds only for linear functions.

The integral inequalities considered here have obvious summation analogues; their proofs can be obtained either by passing to the limit in the summation inequality, or by modifying the proofs used in the discrete case. If a continuously differentiable function $f$ is nondecreasing and convex (or nonincreasing and concave) on the interval $\left[0,1\right]$, then

For a nondecreasing concave (or nonincreasing convex) function the signs of the inequalities are reversed. Equality is possible only for a linear function.

The integral inequalities considered here have obvious summation analogues; their proofs can be obtained either by passing to the limit in the summation inequality, or by modifying the proofs used in the discrete case. Let the functions $f$ and $g$ be positive and integrable on the interval $\left[a, b\right]$. Then the following inequalities hold:

The integral inequality considered here has an obvious summation analogue; its proof can be obtained either by passing to the limit in the summation inequality, or by modifying the proof used in the discrete case. Let the functions $f$ and $g$ be monotonic on the interval $\left[a, b\right]$, and let the function $h$ be nonnegative and integrable on it. If the monotonicity of the functions $f, g$ is of the same type, then the following inequality holds:

while if the monotonicity is of different types, then the sign of the inequality is reversed.

1. Let $f \in C^{1}(\left[0, \pi \right])$, with $f(0)=f(\pi )=0$. Prove the inequality

   $$\int _{0}^{\pi } f^{2}(x) d x \leqslant \int _{0}^{\pi }\left(f^{\prime }(x)\right)^{2} d x$$

   by representing the difference of these integrals as an integral of a nonnegative function.

2. Prove that the inequality remains valid if, instead of the condition $f(0)=f(\pi )=0$, we assume that $f(0)=f(\pi )$ and $\int_{0}^{\pi } f(x) d x=0$.

3. Let $f \in C^{1}(\left[0,2 \pi \right]), f(0)=f(2 \pi )$ and $\int_{0}^{2 \pi } f(x) d x=0$. Prove that

   $$\int _{0}^{2 \pi } f^{2}(x) d x \leqslant \int _{0}^{2 \pi }\left(f^{\prime }(x)\right)^{2} d x$$

For which functions do these inequalities become equalities?

Let a smooth closed curve of length $L$ bound a plane figure of area $S$. Using the natural parametrization of the curve and Wirtinger's inequality, prove the isoperimetric inequality: $4 \pi S \leqslant L^{2}$. Use it to determine the maximal area of a figure that can be bounded by a curve of length $L$, lying in a half-plane, with endpoints on the boundary of the half-plane — "Dido's problem"${ }^{*)}$.