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On univalent solutions of the biharmonic equation. (English) Zbl 1100.30006

The biharmonic equation

ΔΔu=0

occurs in many physical situations, most important applications are in fluid dynamics and elasticity problems [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 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 2 G+H,z=re iϕ , 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 2 G(z),|z|<1, where G is harmonic, then F is starlike whenever G is starlike.


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