IMO 1967 LL USS58

Origin: USS

Problem

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 \leqx \leq1 (y = 0) of the real line in the complex plane (z = x + iy) is equal to M. Prove that for every z |l(z)| \leqM\rho, where \rho is the sum of distances from the point P = z to the points Q1: z = 1 and Q3: z = −1.

Solution

The following sequence of equalities and inequalities gives an even stronger estimate than needed. |l(z)| = |Az + B| = 1 2|(z + 1)(A + B) + (z −1)(A −B)| = 1 2|(z + 1)f(1) + (z −1)f(−1)| \leq1 2(|z + 1| \cdot |f(1)| + |z −1| \cdot |f(−1)|) \leq1 2(|z + 1| + |z −1|)M = 1 2\rhoM.