This paper deals with applications of a new approach to symmetrization which is called polarization in order to derive new inequalities for the Green’s function, harmonic measures and the Poincaré metric. The concept of polarization was introduced for plane sets by V. Wolontis [Am. J. Math. 74, 587-606 (1952; Zbl 0047.08001)] and for functions by A. Baernstein and B. A. Taylor [Duke Math. J. 43, 245-268 (1976; Zbl 0331.31002)] and later by V. N. Dubinin [Math. Zametki 38, 49-55 (1985; Zbl 0603.31001)].
Let \(H\) be a hyperplane in \(\overline{\mathbb{R}}^n\), \(x^*(H)\) a point in \(\overline{\mathbb{R}}^n\) which is symmetric to \(x\) with respect to \(H\), \(A^*(H)=\{x:x^*\in A\}\) and \(A^\pm=A\cap (\overline{\mathbb{R}}^n)^\pm\). Then the polarized set \(P_HA\) of \(A\) with respect to \(H\) is defined in the following way: \(P_HA=(A\cup A^*)^+\cup(A\cap A^*)^-\).
A typical result is the following: Let \(D\) be a domain in \(\mathbb{R}^n\) and \(D_H\) the simply connected component of \(P_HD\). \(G(x,y)\) denotes the Green’s function of the domain \(G\) and \(G_H\) denotes the Green’s function of the domain \(G_H\). The author proves
Theorem 1. For \(x,y\in (D_H)^+\) \[ \begin{aligned} G_H(x,y) & \geq \max\{G(x,y),G(x,y^*),G(x^*,y),G(x^*,y^*)\},\\ G_H(x,y) + G_H(x^*,y) & \geq \max\{G(x,y) + G(x^*,y),G(x,y^*) + G(x^*,y^*)\},\\ G_H(x,y) - G_H(x^*,y) & \geq \max\{|G(x,y) - G(x^*,y)|, |G(x,y^*) - G(x^*,y^*)|\}.\end{aligned} \] Equality holds if and only if \(G_H(x,y)\equiv G(x,y)\) or \(G_H(x,y)\equiv G(x^*,y^*)\).