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The extremal limit is univalent
Statement
In the setting of the extremal problem, any holomorphic limit attaining the supremal derivative is univalent.
Facts & Assumptions
Given: A locally uniformly convergent maximizing subsequence from the extremal family, with limit .
The limit satisfies (A maximizing sequence has a locally uniform limit with extremal derivative).
A nonconstant locally uniform limit of univalent functions is univalent (A nonconstant locally uniform limit of univalent functions is univalent).
Proof
By [L1], the derivative of at the basepoint is positive, so is nonconstant.
The maximizing sequence consists of univalent maps, so [L2] applies to the local uniform convergence in the given data. Together with step 1.1 it yields that is univalent.
Depends on
Used by
Dependency tree · two levels
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Sources
- Matthias Weber, Complex Analysis, Theorem 5.2.6 (standard reference, not scraped)
- Walter Rudin, Real and Complex Analysis, Theorem 14.9 (standard reference, not scraped)