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On some properties of solutions of the biharmonic equation. (English) Zbl 1098.31002
It is shown that the complex linear univalent operator $L(f) \equiv z f_{z} - \overline{z} f_{\overline{z}}$ preserves harmonicity and biharmonicity in the unit disc. It is proved that for a biharmonic mapping in the unit disc having the form $F = r^2 G$ with $G$ being harmonic and orientation preserving the Jacobian $J_F = \vert F_z\vert ^2 - \vert F_{\overline{z}}\vert ^2$ is positive for all $0 < r < 1$ and $J_F(0) = 0$. Starlikeness property of functions of the form $F = r^2 G$ with $G$ being harmonic and orientation preserving is characterized too.

MSC:
31A30Biharmonic (etc.) functions and equations (two-dimensional), Poisson’s equation
31A05Harmonic, subharmonic, superharmonic functions (two-dimensional)
30C45Special classes of univalent and multivalent functions
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References:
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