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Landau’s theorem for biharmonic mappings. (English) Zbl 1156.31001
The article ‘Landau’s theorem for biharmonic mappings’, having as authors Z. Abdulhadi and Y. Abu Muhanna, contains: - Introduction - Schwarz lemma - Theorem 1 and Theorem 2 of the authors. The Introduction states the necessary and sufficient condition that the continuously differentiable complex-valued function must be biharmonic in a simply connected domain D, using an article of S. Khuri and the authors [J. Inequal. Appl. 2005, No. 5, 469–478 (2005; Zbl 1100.30006)]. This theorem reduces the discussion of the ‘biharmonic mapping’ condition to harmonic functions (or to a certain linear combination of harmonic functions). The Schwarz lemma determines the boundary of function f a harmonic mapping of the unit disk. Theorem 1 and Theorem 2 of the present article’s authors show that Landau’s theorem extends to the bounded biharmonic mapping of the unit disk. Thus, Theorem 1 shows that if is a biharmonic mapping in (unit disk), boundary conditions take place for the module of harmonic functions and , where the function is supposed to be univalent in a disk with ray . The size of ray depends on the boundary of the module of functions. Theorem 2 only considers function having the module bounded by and univalent in the disk with ray. The equation which determines the size of ray is cubic and it depends on boundary. The demonstrations for Theorem 1 and Theorem 2 consider functions and to be analytical and use their developing into power series in the interior of a disk completely included into the unit disk. The variation of function in both theorems is calculated using, the line-segment in the disk. The authors state at every given theorem that: ‘this result is not sharp’.

MSC:
31A05Harmonic, subharmonic, superharmonic functions (two-dimensional)
References:
[1]Abdulhadi, Z.; Muhanna, Y. Abu; Khoury, S.: On univalent solutions of the biharmonic equations, J. inequal. Appl. (5), 469-478 (2005) · Zbl 1100.30006 · doi:10.1155/JIA.2005.469
[2]Muhanna, Y. Abu; Schober, G.: Harmonic mappings onto convex mapping domains, Canad. J. Math. 39, No. 6, 1489-1530 (1987) · Zbl 0644.30003 · doi:10.4153/CJM-1987-071-4
[3]Chen, H.; Gauthier, P.; Hengartner, W.: Bloch constants for planar harmonic mappings, Proc. amer. Math. soc. 128, 3231-3240 (2000) · Zbl 0956.30012 · doi:10.1090/S0002-9939-00-05590-8
[4]Clunie, J.; Sheil-Small, T.: Harmonic univalent functions, Ann. acad. Sci. fenn. Math. ser. A 9, 3-25 (1984) · Zbl 0506.30007
[5]Dorff, M.; Nowark, M.: Landau’s theorem for planar harmonic mappings, Comput. methods funct. Theory 4, No. 1, 151-158 (2004) · Zbl 1060.30033
[6]Duren, P.: Harmonic mappings in the plane, (2004)
[7]Grigoryan, A.: Landau and Bloch theorems for harmonic mappings, Complex var. Theory appl. 51, No. 1, 81-87 (2006) · Zbl 1114.30024 · doi:10.1080/02781070500369214