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Zbl 1098.31002
Abdulhadi, Z.; Muhanna, Y.Abu; Khuri, S.
On some properties of solutions of the biharmonic equation.
(English)
[J] Appl. Math. Comput. 177, No. 1, 346-351 (2006). ISSN 0096-3003

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.
[Sergei V. Rogosin (Minsk)]
MSC 2000:
*31A30 Biharmonic (etc.) functions and equations (two-dim.)
31A05 Harmonic functions, etc. (two-dimensional)
30C45 Special classes of univalent and multivalent functions

Keywords: harmonic functions; biharmonic functions; univalent functions; starlike domains; complex linear differential operator

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