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- W2279254075 abstract "Loewner’s differential equation [7] $$tfrac{partial } {{partial {kern 1pt} t}}fleft( {t,z} right) = phi left( {t,z} right)zfrac{partial } {{partial {kern 1pt} z}}fleft( {t,z} right) $$(1)arises in the estimation theory of Riemann mapping functions. We consider (1) in a vector setting with operator valued Herglotz functions Φ(t,z). Notions of geometric function theory are lost in this generality, but the intuitive idea of an expansive flow remains. Mathematically, it is more convenient to think of the flow as contractive in reverse time. In the scalar case, a classical subordination theorem [9] implies that the evolution transformations for (1), T(a,b): f(b,z) → f(a,z), are contractive in the Dirichlet space if a≤b. For different reasons, a similar result holds in the vector extension. The proof of the Bieberbach conjecture [2] and its power extensions [3], [11] yield other examples. As an illustration of the vector theory, we discuss a result from [11] which includes and extends the example of the Dirichlet space." @default.
- W2279254075 created "2016-06-24" @default.
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- W2279254075 date "1990-01-01" @default.
- W2279254075 modified "2023-09-25" @default.
- W2279254075 title "A Vector Extension of Loewner’s Differential Equation" @default.
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- W2279254075 doi "https://doi.org/10.1007/978-3-0348-7250-8_22" @default.
- W2279254075 hasPublicationYear "1990" @default.
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