Inequalities for polynomials in the complex plane

Let $K$ be a compact, connected subset of $\mathbb {C}$ containing more than one point and let $f$ be a one-to-one conformal map $\mathbb {C} \setminus K$ onto $\Delta = \left\{ z : \left|z\right| < 1\right\}$ with $f(\infty ) = 0$. If $p$ is a polynomial of degree $n$ for which $\left|p(z)\right| \leq 1$ for $z \in K$, prove that

Prove the theorem: if a polynomial $f(z)$ of degree $n$ does not exceed $M$ in modulus on the interval $-1 \leqslant z \leqslant 1$, then for any $z$ the inequality $\left|f(z)\right| \leqslant M(a+b)^{n}$ holds, where $a$ and $b$ are the semi-axes of the ellipse passing through the point $z$ and having its foci at the points $z = \pm 1$.

Suppose that $f(z)$ is a polynomial of degree $n$ and that

on the real interval $-1\leq z\leq 1$. Then we have

for any $z$ outside this interval; $a$ and $b$ are the semi-axes of the ellipse through $z$ and with foci $-1$ and $1$. What does the proposition imply for $z\to \infty$?

We assume that $f(z)$ is a polynomial of degree $n$, that $E_{1}$ and $E_{2}$ denote two homofocal ellipses with semi-axes $a_{1},b_{1}$ and $a_{2},b_{2}$, $a_{1}<a_{2}$, $b_{1}<b_{2}$. The maximum of $\left|f(z)\right|$ on $E_{1}$ and $E_{2}$ is denoted by $M_{1}$ and $M_{2}$ resp.; then

Derive problems 269 and 270 from this proposition.

Let $P(z)$ be a polynomial of degree $n$, and let $M(r) = \max_{\left|z\right| = r}\left|P(z)\right|$. Prove that for $0 < r_{1} < r_{2}$ the inequality

holds, and equality, for even one pair of values $r_{1}$ and $r_{2}$, is possible only for a polynomial of the form $P(z) = a z^{n}$.

Let $P(z) = z^{n}+a_{1} z^{n-1}+\ldots +a_{n}$. Prove that either the inequality $\left|P(z)\right| > 1$ holds at some point of the circle $\left|z\right| = 1$, or $P(z) \equiv z^{n}$.

Let $P(z)$ be a polynomial of degree $n$ satisfying, on the interval $(-1,1)$, the inequality $\left|P(z)\right| \leqslant M$. Prove that at every point $z_{0}$ lying outside this interval the inequality $\left|P\left(z_{0}\right)\right| \leqslant M(a+b)^{n}$ holds, where $a$ and $b$ are the semi-axes of the ellipse with foci $-1$ and $1$ passing through the point $z_{0}$.