We prove that $$\max_{ t \in [-\pi,\pi]}{ |Q(t)|} \leq T_{ 2n}(\sec(s/4)) = \frac 12 ((\sec(s/4) + \tan(s/4))^{ 2n} + (\sec(s/4) - \tan(s/4))^{ 2n})$$ for every even trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m(\{ t \in [-\pi,\pi]: |Q(t)| \leq 1\}) \geq 2\pi-s\,, \qquad s \in (0,2\pi)\,,$$ where $m(A)$ denotes the Lebesgue measure of a measurable set $A \subset { \Bbb R}$ and $T_{ 2n}$ is the Chebysev polynomial of degree $2n$ on $[-1,1]$ defined by $T_{ 2n}(\cos t) = \cos(2nt)$ for $t \in { \Bbb R}$. This inequality is sharp. We also prove that $$\max_{ t \in [-\pi,\pi]}{ |Q(t)|} \leq T_{ 2n}(\sec(s/2)) = \frac 12 ((\sec(s/2) + \tan(s/2))^{ 2n} + (\sec(s/2) - \tan(s/2))^{ 2n})$$ for every trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m(\{ t \in [-\pi,\pi]: |Q(t)| \leq 1\}) \geq 2\pi-s\,, \qquad s \in (0,\pi)\,.$$
Categories: math.CA