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