  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
sharp 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. 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 $$ \int_{0}^{2\pi}
|f(re^{i\theta})|^{p}d\theta\le\int_{0}^{2\pi}
|G(re^{i\theta})|^{p}d\theta,\,\,0<p<\infty $$ which is true for $p=2$ (we
prove) but can not be true for too large $p$. What is the largest such exponent
or its supremum?
