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On univalent solutions of the biharmonic equation. (English) Zbl 1100.30006
The biharmonic equation $$ \Delta \Delta u = 0 $$ occurs in many physical situations, most important applications are in fluid dynamics and elasticity problems [{\it St. Bergman, M. M. Schiffer}, Kernel Functions and Elliptic Differential Equations, (Academic Press, New York) (1953; Zbl 0053.39003)]. A continuous complex-valued function $F=u+iv$ in a domain $D \subset \mathbb{C}$ is biharmonic if $F$ satisfies the biharmonic equation. The paper deals with univalent and starlike biharmonic functions. If $F$ is biharmonic in any simply connected domain then it holds $F=r^2G+H,z=re^{i\varphi},$ where $G$ and $H$ are harmonic. It is proven for example that if $F$ is biharmonic in the unit disk and it holds $F(z) = r^2G(z), \vert z\vert <1,$ where $G$ is harmonic, then $F$ is starlike whenever $G$ is starlike.

30C45Special classes of univalent and multivalent functions
31A30Biharmonic (etc.) functions and equations (two-dimensional), Poisson’s equation
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