Quantum field theory and the Bieberbach conjecture

An intriguing correspondence between ingredients in geometric function theory related to the famous Bieberbach conjecture (de Branges’ theorem) and the non-perturbative crossing symmetric representation of 2-2 scattering amplitudes of identical scalars is pointed out. Using the dispersion relation and unitarity, we are able to derive several inequalities, analogous to those which arise in the discussions of the Bieberbach conjecture. We derive new and strong bounds on the ratio of certain Wilson coefficients and demonstrate that these are obeyed in one-loop \phi^4ϕ4 theory, tree level string theory as well as in the S-matrix bootstrap. Further, we find two sided bounds on the magnitude of the scattering amplitude, which are shown to be respected in all the contexts mentioned above. Translated to the usual Mandelstam variables, for large |s||s|, fixed tt, the upper bound reads |\mathcal{M}(s,t)|\lesssim |s^2||ℳ(s,t)|≲|s2|. We discuss how Szeg"{o}’s theorem corresponds to a check of univalence in an EFT expansion, while how the Grunsky inequalities translate into nontrivial, nonlinear inequalities on the Wilson coefficients.

A note on successive coefficients of spirallike functions

Even there were several facts to show that ||an+1(f)|-|an(f)|| ? 1 is not true for the whole class of normalised univalent functions in the unit disk with with the form f(z) = z + ??,k=2 akzk. In 1978, Leung[7] proved ||an+1(f)|-|an(f)|| is actually bounded by 1 for starlike functions and by this result it is easy to get the conclusion |an| ? n for starlike functions. Since ||an+1(f)|-|an(f)|| ? 1 implies the Bieberbach conjecture (now the de Brange theorem), so it is still interesting to investigate the bound of ||an+1(f)|-|an(f)|| for the class of spirallike functions as this class of functions is closely related to starlike functions. In this article we prove that this functional is bounded by 1 and equality occurs only for the starlike case. We are also able to give a precise form of extremal functions. Furthermore we also try to find the sharp bound of ||an+1(f)|-|an(f)|| for non-starlike spirallike functions. By using the Carath?odory-Toeplitz theorem, we obtain the sharp lower and upper bounds of |an+1(f)|-|an(f)| for n = 1 and n = 2. These results disprove the expected inequality ||an+1(f)|-|an(f)||? cos ? for ?-spirallike functions.

A certain class of q-close-to-convex functions of order α

For every 0 < q < 1 and 0 ? ? < 1, we introduce a class of analytic functions f on the open unit disc D with the standard normalization f(0)= 0 = f'(0)-1 and satisfying |1/1-?(z(Dqf)(z)/h(z)-?)- 1/1-q,(z?D), where h?S*q. This class is denoted by Kq(?), so called the class of q-close-to-convex-functions of order ?. In this paper, we study some geometric properties of this class. In addition, we consider the famous Bieberbach conjecture problem on coefficients for the class Kq(?). We also find some sufficient conditions for the function to be in Kq(?) for some particular choices of the functions h. Finally, we provide some applications on q-analogue of Gaussian hypergeometric function.

Proof of the Zalcman conjecture for initial coefficients

The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions on the unit disk satisfy the inequality for all n > 2, with the equality only for the Koebe function. This conjecture remained open for n > 3. We provide here the proof of this inequality for n = 4, 5, 6. It relies on the holomorphic homotopy of univalent functions and comparison of generated singular conformal metrics in the disk. The extremality of Koebe's function follows from hyperbolic properties.