Sub-Riemannian geometry of the coefficients of univalent functions

Sub-riemannian Geometry of the Coefficients of Univalent Functions

We consider coefficient bodies Mn for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov’s operators for a representation of the Virasoro algebra. Then Mn are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace a...

Coefficients of Univalent Functions

The interplay of geometry and analysis is perhaps the most fascinating aspect of complex function theory. The theory of univalent functions is concerned primarily with such relations between analytic structure and geometric behavior. A function is said to be univalent (or schlichi) if it never takes the same value twice: f(z{) # f(z2) if zx #= z2. The present survey will focus upon the class S ...

Bounds for the Coefficients of Univalent Functions

assumed regular and Univalent in \z\ <1, in terms of the domain onto which \z\ <1 is mapped through (1). A typical result, cf. (27), is that if this domain does not cover arbitrarily large circles, then1 a„ = 0(log n). Let W be the domain in the w-plane onto which \z\ <1 is mapped through (1) and denote by .4(72) the radius of the largest circle with center on |w| = 72 the whole interior of whi...

Sub - Riemannian geometry :

Take an n-dimensional manifold M . Endow it with a distribution, by which I mean a smooth linear subbundle D ⊂ TM of its tangent bundle TM . So, for x ∈ M , we have a k-plane Dx ⊂ TxM , and by letting x vary we obtain a smoothly varying family of k-planes on M . Put a smoothly varying family g of inner products on each k-plane. The data (M,D, g) is, by definition, a sub-Riemannian geometry. Tak...

Sub-riemannian Geometry