This chapter marks the debut of the star function in the book. Each type of rearrangement has an associated star function, which is an indefinite integral of the rearranged function. This chapter proves ``subharmonicity'' theorems for the star function, expressing the fact that if a function satisfies a Poisson-type partial differential equation then its star function satisfies a related differential inequality. In the simplest case of circular symmetrization in the plane, the result says that if a function is subharmonic then so is its star function. Subharmonicity is applied in the succeeding chapters to yield comparison theorems for solutions of partial differential equations and extremal results in complex analysis.

Chapter 6 discusses Steiner symmetrization. Basic properties of symmetric decreasing rearrangement and polarization that were developed in Chapter 1 are adapted to Steiner symmetrization, to show that it decreases the modulus of continuity and acts contractively in L-Infinity.The effect of Steiner symmetrization on various Dirichlet integrals is studied. It is shown that Steiner symmetrization decreases perimeter and Minkowski content, but in general it is not known whether it decreases the (n-1)-dimensional Hausdorff measure. Steiner symmetrization also decreases the principal frequency andvarious capacities, and increases the torsional rigidity and mean lifetime of a Brownian particle.