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Symmetrization and the Poincaré metric. (English) Zbl 0604.30028

Let D be a plane domain, and denote by \(D^*\) its circular symmetrization. The author proves a famous conjecture by establishing the inequality \[ \lambda_{D^*}(r)\leq \sup_{| z| =r}\lambda_ D(z) \] where \(\lambda_ D\), \(\lambda_{D^*}\) denote the corresponding Poincaré metrics. A corollary is that Hayman’s ”principle of symmetrization” holds for arbitrary D: If f is analytic in the unit disc \(\Delta\), with \(f(\Delta)\subset D\), then \(| f'(0)| \leq F'(0)\), \(M(r,f)\leq M(r,f)\), where F is the universal covering map of \(\Delta\) onto \(D^*\) with \(f(0)=| f(0)|\), F’(0)\(\geq 0\). Hayman (1950), had shown this is true when \(D^*\) is simply connected.
The author’s P.M. inequality is a special case of a general symmetrization-comparison theorem he proves for solutions of equations of the form \(\Delta u=g(u)\), where \(g\in C^ 2({\mathbb{R}})\) is convex and increasing. His main tool is the theory of ”*-functions”, introduced in the early 1970’s by the reviewer. The fact that this theory can be successfully applied to nonlinear problems is very gratifying, and opens up many new possibilities.
Reviewer: A.Baernstein II

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
35J60 Nonlinear elliptic equations
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