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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:
 31A30 Biharmonic (etc.) functions and equations (two-dimensional), Poisson’s equation 31A05 Harmonic, subharmonic, superharmonic functions (two-dimensional) 30C45 Special classes of univalent and multivalent functions
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##### References:
 [1] Abu-Muhanna, Y.; Schober, G.: Harmonic mappings onto convex mapping domains. Can. J. Math 39, No. 6, 1489-1530 (1987) · Zbl 0644.30003 [2] Clunie, J.; Sheil-Small, T.: Harmonic univalent functions. Ann. acad. Sci. fenn. Series A. Math. 9, 3-25 (1984) · Zbl 0506.30007 [3] Choquet, G.: Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques. Bull. sci. Math. 69, No. 2, 51-60 (1945) [4] Pommerenke, C.: Univalent functions. (1975) · Zbl 0298.30014 [5] Happel; Brenner, H.: Low Reynolds number hydrodynamics. (1965) · Zbl 0612.76032 [6] Khuri, S. A.: Biorthogonal series solution of Stokes flow problems in sectorial regions. SIAM J. Appl. math. 56, No. 1, 19-39 (1996) · Zbl 0844.76019 [7] Langlois, W. E.: Slow viscous flow. (1964) [8] Abdulhadi, Z.; Abu-Muhanna, Y.; Khuri, S.: On the univalence of log-biharmonic mappings. J. math. Anal. appl. 289, 629-638 (2004) · Zbl 1036.30007 [9] Abdulhadi, Z.; Abu-Muhanna, Y.; Khuri, S.: On univalent solutions of the biharmonic equation. J. inequal. Appl. 2005, No. 5, 469-478 (2005) · Zbl 1100.30006