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Let p(z) be a polynomial of the form $p\left(z\right)={\sum }_{j=0}^n{a}_j{z}^j$, $a_j\in -1,1$. We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.