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