The Theorems of S. Bernstein and A. Markov

For the set of all polynomials of degree $n-1$, $P(x) = a_0 x^{n-1} + a_1 x^{n-2} + \cdots + a_{n-1}$, which satisfy in the interval $-1 \leqq x \leqq 1$ the inequality

the maximum value that $\left|a_0\right|$ can assume is equal to $2^{n-1}$, i.e.

The equality sign holds if and only if $P(x) = \gamma U_{n-1}(x)$, $\left|\gamma \right| = 1$.

Deduce from problem 73 a new proof of problem 62.

Deduce from problem 74 a new proof of problem 65.

Consider the set of all polynomials $P(x)$ of degree $n-1$ which satisfy in the interval $-1 \leqq x \leqq 1$ the inequality

Then we have

The equality sign holds only if $P(x) = \gamma U_{n-1}(x)$ with $\left|\gamma \right|=1$ and $x = \pm 1$.

Prove the following generalization of the second inequality in problem 7: Let

be an arbitrary sine polynomial of $n$th order with real coefficients for which we have $\left|S(\vartheta )\right| \leqq 1$. Then

The equality sign holds only for $S(\vartheta ) = \pm \sin n\vartheta$.

Let the trigonometric polynomial of $n$th order with real coefficients

satisfy for all real $\vartheta$ the inequality

Then we have

Let the polynomial $P(x)$ of $n$th degree satisfy in the interval $-1 \leqq x \leqq 1$ the inequality $\left|P(x)\right| \leqq 1$. Then we have

The equality sign holds only if $P(x) = \gamma T_n(x)$, $\left|\gamma \right|=1$, $x = \pm 1$.