A new approach in Loewner Theory proposed by Bracci, Contreras, Díaz-Madrigal and Gumenyuk provides a unified treatment of the radial and the chordal versions of the Loewner equations. In this framework, a generalized Loewner chain satisfies the differential equation

where τ: [0, ∞) → \(\overline{\mathbb{D}}\) is measurable and p is called a Herglotz function. In this paper, we will show that if there exists a k ⩽ [0, 1) such that p satisfies

for all z ⩽ \(\mathbb{D}\) and almost all t ⩽ [0, ∞), then, for all t ⩽ [0, ∞), f_t has a k-quasiconformal extension to the whole Riemann sphere. The radial case (τ = 0) and the chordal case (τ = 1) have been proven by Becker [J. Reine Angew. Math., vol. 255 (1972), 23–43] and Gumenyuk and the author [Math. Z., vol. 285 (2017), 1063–1089]. In our theorem, no superfluous assumption is imposed on τ ⩽ \(\overline{\mathbb{D}}\). As a key foundation of the proof is an approximation method using a continuous dependence of evolution families and Loewner chains.

Bracci，Contreras，Díaz-Madrigal和Gumenyuk提出的一种新的Loewner理论方法提供了对Loewner方程的径向和弦形式的统一处理。在此框架中，广义的Loewner链满足微分方程

其中τ：[0，∞）→ \（\ overline {\ mathbb {D}} \）是可测量的，并且p称为Herglotz函数。在本文中，我们将证明，如果存在一个ķ ⩽[0，1），使得p满足

对于所有ž ⩽ \（\ mathbb {d} \）并且几乎所有吨⩽[0，∞），那么，对于所有吨⩽[0，∞），˚F_吨具有ķ -quasiconformal扩展到整个黎曼球体。径向情况（τ= 0）和弦情况（τ= 1）已由Becker证明[J。雷尼·安格（Reine Angew）。数学卷 255（1972），23-43]和Gumenyuk及其作者[Math。Z.，第一卷 285（2017），1063-1089]。在我们的定理中，τ⩽ \（\ overline {\ mathbb {D}} \）上没有多余的假设。作为证明的关键基础，是一种使用进化族和Loewner链的连续依赖性的近似方法。