AUTHORS: Ronen Peretz
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ABSTRACT: In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some inequalities on analytic functions in the unit disk. We answer a question that was asked by Albert Baernstein II, regarding the coefficients of circular symmetrization functions. We mostly deal with the Steiner symmetrization G of an analytic function f in the unit disk U. We pose few problems we can not solve. An intriguing one is that of the inequality Z2π 0 |f(reiθ)| p dθ ≤ Z2π 0 |G(reiθ)| p dθ, 0 < p < ∞ which is true for p = 2 but can not be true for too large p. What is the largest such exponent or its supremum?
KEYWORDS: circular symmetrization, Steiner symmetrization, extremal problems.
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