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The main result says in particular that if $t\left(\zeta \right):={\sum }_{\nu =-n}ⁿc_{\nu }{e}^{i\nu \zeta }$ is a trigonometric polynomial of degree n having all its zeros in the open upper half-plane such that |t(ξ)| ≥ μ on the real axis and cₙ ≠ 0, then |t’(ξ)| ≥ μn for all real ξ.