Steiner symmetrization and the initial coefficients of univalent functions

摘要

We establish the inequality |a_1|~2 - Rea _1a-1 ≥ |a1*|~2 - Rea1*a*_(-1) for the initial coefficients of any function f (z) = a_(1z+a0) +a _(-1)/z+ ? ? ? meromorphic and univalent in the domain D = {z: |z| > 1}, where a*_1 and a*_(-1) are the corresponding coefficients in the expansion of the function f *(z) that maps the domain D conformally and univalently onto the exterior of the result of the Steiner symmetrization with respect to the real axis of the complement of the set f (D). The Pólya-Szego inequality |a_1| ≥ |a*_1| is already known. We describe some applications of our inequality to functions of class ∑.