Question 58
(USS 5) A linear binomial $l(z)=Az+B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1\le x\le1$ $(y=0)$ of the real line in the complex plane $(z=x+iy)$ is equal to $M$. Prove that for every $z$
$$|l(z)|\le M\rho,$$
where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1$: $z=1$ and $Q_3$: $z=-1$.
$$|l(z)|\le M\rho,$$
where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1$: $z=1$ and $Q_3$: $z=-1$.