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Landau’s theorem for certain biharmonic mappings. (English) Zbl 1159.31301

Summary: We show the existence of Landau and Bloch constants for biharmonic mappings of the form \(L(F)\). Here \(L\) represents the linear complex operator \(L = z\frac{\partial }{\partial z}-\bar z\frac{\partial }{\partial z} \) defined on the class of complex-valued \(C^{1}\) functions in the plane, and \(F\) belongs to the class of biharmonic mappings of the form \(F(z)=|z|^{2}G(z)+K(z)\) \((|z|<1)\), where \(G\) and \(K\) are harmonic.

MSC:

31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
30C62 Quasiconformal mappings in the complex plane
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