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A new version of the symmetrization theorem is proved. Using a modification of the «-function of Baernstein we construct an operator which maps a family of ¿-subharmonic functions defined on an annulus into a family of subharmonic functions on an annular sector. Applying this operator to the Green function of special domains we prove A. Weitsman's conjecture linked with exact estimates of the Green functions of these domains.doi:10.2307/2154986 fatcat:uqfz6ajgfnhnfaxvzf3aiyirjq