×

zbMATH — the first resource for mathematics

On the univalence of functions with logharmonic Laplacian. (English) Zbl 1179.31001
Summary: We prove that Radó’s theorem holds for functions of the form \(F(z) = r^2L(z)\), where \(L\) is logharmonic. We show that if \(F\) is of the form \(F(z) = r^2L(z)\), \( |z| < 1\), where \(L(z) = h(z)\overline {g(z)}\) is logharmonic, then \(F\) is starlike iff \(\psi (z) = h(z)/g(z)\) is starlike. In addition, when \(F(z) = r^2L(z) + H(z)\), \(|z| < 1\), where \(L\) is logharmonic and \(H\) is harmonic, we give sufficient conditions for \(F\) to be locally univalent.

MSC:
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abdulhadi, Z., Close-to-starlike logharmonic mappings, Int. J. math. math. sci., 19, 3, 563-574, (1996) · Zbl 0921.30014
[2] Abdulhadi, Z., Typically real logharmonic mappings, Int. J. math. math. sci., 31, 1, 1-9, (2002) · Zbl 1010.30016
[3] Abdulhadi, Z.; Bshouty, D., Univalent functions in \(H \overline{H}\), Trans. am. math. soc., 305, 2, 841-849, (1988) · Zbl 0661.30017
[4] Abdulhadi, Z.; Hengartner, W., Spirallike logharmonic mappings, Complex variable theory appl., 9, 2-3, 121-130, (1987) · Zbl 0643.30011
[5] Abdulhadi, Z.; Hengartner, W., One pointed univalent logharmonic mappings, J. math. anal. appl., 203, 2, 333-351, (1996) · Zbl 0864.30037
[6] Abu-Muhanna, Y.; Schober, G., Harmonic mappings onto convex mapping domains, Can. J. math., XXXIX, 6, 1489-1530, (1987) · Zbl 0644.30003
[7] Clunie, J.; Sheil-Small, T., Harmonic univalent functions, Ann. acad. sci. fenn. ser. A: math., 9, 3-25, (1984) · Zbl 0506.30007
[8] Choquet, G., Sur un type de transformation analytique géné ralisant la représentation conforme et définie au moyen de fonctions harmoniques, Bull. sci. math., 69, 2, 51-60, (1945)
[9] Goodman, A.W., Univalent functions, vol. I, (1983), Mariner Publishing Company INS · Zbl 1041.30500
[10] T. Radó, Aufgabe 41, Jahresber. Deutsch Math. Verein 35 (1926) 49.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.