Univalence and holomorphic extension of the solution to $\omega$-controlled Loewner-Kufarev equations

TL;DR

This paper proves that solutions to the ${ ext{omega}}$-controlled Loewner-Kufarev equations are unique, univalent, starlike within the unit disk, and can be holomorphically extended beyond the boundary.

Contribution

It establishes the existence, uniqueness, and holomorphic extendability of solutions to the ${ ext{omega}}$-controlled Loewner-Kufarev equations, advancing understanding of their geometric properties.

Findings

Solutions are unique and univalent within the unit disk.

Solutions are starlike on the unit disk.

Solutions can be extended holomorphically across the boundary.

Abstract

We prove that a solution to the $\omega$-controlled Loewner-Kufarev equation, which was introduced by the first two authors, exists uniquely, is univalent and starlike on the unit disk and can be extended holomorphically across the unit circle.