Working with Bombieri Inequality part8(Machine Learning 2024) – Monodeep Mukherjee

  1. Optimal control in Bombieri’s and Tammi’s conjectures(arXiv)

Authors: Dmitri Prokhorov, Alexander Vasil’ev

Abstract: Let S stand for the usual class of univalent regular functions in the unit disk U={z:|z|<1} normalized by f(z)=z+a2z2+… in U, and let SM be its subclass defined by restricting |f(z)|<M in U, M≥1. We consider two classical problems: Bombieri’s coefficient problem for the class S and the sharp estimate of the fourth coefficient of a function from SM. Using Löwner’s parametric representation and the optimal control method we give exact initial Bombieri’s numbers and derive a sharp constant M0, such that for all M≥M0 the Pick function gives the local maximum to |a4|. Numerical approximation is given.

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