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 $Q_3: z = -1.$
Darij Grinberg's translation from Morozova/Petrakov:
Let $z$ denote a variable complex number. Given the linear function $l\left( z\right) =Az+B$ with complex coefficients $A$ and $B.$ Given that the maximum of $\left| l\left( z\right) \right| $ on the interval $-1\leq x\leq 1$ (with $y=0$) of the real number axis of the complex plane $ z=x+iy$ is $M.$ Show that for any $z,$ the inequality $\left| l\left( z\right) \right| \leq M\rho $ holds, where $\rho $ is the sum of the distances from the point $P=z$ to the points $Q_{1}$ (corresponding to the complex numer $1$) and $Q_{3}$ (corresponding to the complex number $-1$).
Please post your solutions. This is just a solution template to write up your solutions in a nice way and formatted in LaTeX. But maybe your solution is so well written that this is not required finally. For more information and instructions regarding the ISL/ILL problems please look here: introduction for the IMO ShortList/LongList project and regardingsolutions