The sharp Remez-type inequality for even trigonometric polynomials on the period

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 $$m({t in [-pi,pi]: |Q(t)| leq 1}) geq 2pi-s,, qquad s in (0,2pi),,$$ 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 $$m({t in [-pi,pi]: |Q(t)| leq 1}) geq 2pi-s,, qquad s in (0,pi),.$$