For bounded convex domains, equality holds in the extremal problem for harmonic maps conformal at a point if and only if the domain is the image of the unit disc under a holomorphic map with derivative of the form c/(1 + a z + λ z²) satisfying the given bounds on a and λ.
Kalaj, A sharp inequality for harmonic diffeomorphisms of the unit disk,J
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On an extremal problem for harmonic maps conformal at a point
For bounded convex domains, equality holds in the extremal problem for harmonic maps conformal at a point if and only if the domain is the image of the unit disc under a holomorphic map with derivative of the form c/(1 + a z + λ z²) satisfying the given bounds on a and λ.