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Properties of a class of \(p\)-harmonic functions. (English) Zbl 1470.30022

Summary: A \(p\) times continuously differentiable complex-valued function \(F = u + i v\) in a domain \(D \subseteq \mathbb{C}\) is \(p\)-harmonic if \(F\) satisfies the \(p\)-harmonic equation \(\Delta \cdots \Delta F = 0\), where \(p\) is a positive integer. By using the generalized Salagean differential operator, we introduce a class of \(p\)-harmonic functions and investigate necessary and sufficient coefficient conditions, distortion bounds, extreme points, and convex combination of the class.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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[1] Clunie, J.; Sheil-Small, T., Harmonic univalent functions, Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, 9, 3-25 (1984) · Zbl 0506.30007
[2] Duren, P., Harmonic Mappings in the Plane. Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, 156, xii+212 (2004), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1055.31001 · doi:10.1017/CBO9780511546600
[3] Avcı, Y.; Złotkiewicz, E., On harmonic univalent mappings, Annales Universitatis Mariae Curie-Skłodowska. Sectio A, 44, 1-7 (1990) · Zbl 0780.30013
[4] Jahangiri, J. M., Harmonic functions starlike in the unit disk, Journal of Mathematical Analysis and Applications, 235, 2, 470-477 (1999) · Zbl 0940.30003 · doi:10.1006/jmaa.1999.6377
[5] Silverman, H., Harmonic univalent functions with negative coefficients, Journal of Mathematical Analysis and Applications, 220, 1, 283-289 (1998) · Zbl 0908.30013 · doi:10.1006/jmaa.1997.5882
[6] Silverman, H.; Silvia, E. M., Subclasses of harmonic univalent functions, New Zealand Journal of Mathematics, 28, 2, 275-284 (1999) · Zbl 0959.30003
[7] Abdulhadi, Z.; Muhanna, Y. A.; Khuri, S., On univalent solutions of the biharmonic equation, Journal of Inequalities and Applications, 2005, 5, 469-478 (2005) · Zbl 1100.30006 · doi:10.1155/JIA.2005.469
[8] AbdulHadi, Z.; Muhanna, Y. A.; Khuri, S., On some properties of solutions of the biharmonic equation, Applied Mathematics and Computation, 177, 1, 346-351 (2006) · Zbl 1098.31002 · doi:10.1016/j.amc.2005.11.013
[9] Abdulhadi, Z.; Abu Muhanna, Y., Landau’s theorem for biharmonic mappings, Journal of Mathematical Analysis and Applications, 338, 1, 705-709 (2008) · Zbl 1156.31001 · doi:10.1016/j.jmaa.2007.05.065
[10] Happel, J.; Brenner, H., Low Reynolds Number Hydrodynamics (1965), New York, NY, USA: Springer, New York, NY, USA
[11] Khuri, S. A., Biorthogonal series solution of Stokes flow problems in sectorial regions, SIAM Journal on Applied Mathematics, 56, 1, 19-39 (1996) · Zbl 0844.76019 · doi:10.1137/0156002
[12] Sălăgean, G. S., Subclasses of univalent functions, Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981). Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), Lecture Notes in Math, 1013, 362-372 (1983), Berlin, Germany: Springer, Berlin, Germany · Zbl 0531.30009 · doi:10.1007/BFb0066543
[13] Al-Oboudi, F. M., On univalent functions defined by a generalized Sǎlǎgean operator, International Journal of Mathematics and Mathematical Sciences, 2004, 27, 1429-1436 (2004) · Zbl 1072.30009 · doi:10.1155/S0161171204108090
[14] Li, S.; Liu, P., A new class of harmonic univalent functions by the generalized Salagean operator, Wuhan University Journal of Natural Sciences, 12, 6, 965-970 (2007) · Zbl 1174.30318 · doi:10.1007/s11859-007-0044-6
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