A general coefficient theorem for univalent functions

יום ב', 08/04/2019 - 14:00

The estimating holomorphic functionals on the classes of univalent functions depending on the
Taylor coefficients $a_n$ of these functions is important in various geometric and physical
applications of complex analysis, because these coefficients reflect the fundamental intrinsic
features of conformal maps.
The goal of the talk is to outline the proof of a new general theorem on maximization of homogeneous
polynomial (in fact, more general holomorphic) coefficient functionals
$$J(f) = J(a_{m_1}, a_{m_2},\dots, a_{m_n})$$
on some classes  of univalent functions in the unit disk naturally connected with the canonical class $S$.
The theorem states that under a natural assumption on zero set of $J$ this functional is maximized only
by the Koebe function $\kappa(z) = z/(1 - z)^2$ composed with pre and post rotations about the origin.
The proof involves a deep result from the Teichm\"{u}ller space theory given by the Bers isomorphism
theorem for Teichm\"{u}ller spaces of punctured Riemann surfaces. The given functional $J$ is lifted
to the Teichm\"{u}ller space $\mathbf T_1$ of the punctured disk $\mathbb D_{*} = \{0 < |z| < 1\}$ which is
biholomorphically equivalent to the Bers fiber space over the universal Teichm\"{u}ller space. This  generates
a positive subharmonic function on the disk $\{|t| < 4\}$ with $\sup_{|t|<4} u(t) = \max_{\mathbf T_1} |J|$
attaining this maximal value only on the boundary circle, which correspond to rotations of the Koebe function.
Our theorem  implies new sharp distortion estimates for univalent functions giving explicitly the extremal
functions  and creates a new bridge between the Teichm\"{u}ller space theory and geometric complex analysis.
In particular, it provides an alternate and direct proof of the Bieberbach conjecture.