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We prove that there are absolute constants $c,C>0$ such that $c/\\log n \\leq \\kappa_n(\\overline{\\mathbb{D}},1) \\leq \\kappa_n(\\mathbb{T},1) \\leq C/\\log n$. Thus the recently established lower bound has the correct order, even when all zeros are required to lie on the unit circle. 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Let $\\kappa_n(K,1)$ denote the least possible area of this set among monic polynomials of degree $n$ whose zeros lie in a compact set $K$. We prove that there are absolute constants $c,C>0$ such that $c/\\log n \\leq \\kappa_n(\\overline{\\mathbb{D}},1) \\leq \\kappa_n(\\mathbb{T},1) \\leq C/\\log n$. Thus the recently established lower bound has the correct order, even when all zeros are required to lie on the unit circle. 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