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Let $\\alpha $ and $\\beta $ be real numbers such that $0\\le \\alpha <\\beta+1$, $\\beta\\le 2 \\alpha$. We show that, among all smooth sets $\\Omega$ in $\\mathbb{R} ^2_+$ with fixed weighted measure $\\iint_{\\Omega } y^{\\beta} dxdy$, the weighted perimeter $\\int_{\\partial \\Omega } y^\\alpha \\, ds$ achieves its minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is explicitly given. 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