×

On the analytic part of univalent harmonic mappings. (English) Zbl 1393.30014

Summary: In this article we obtain two sharp results concerning the analytic part of harmonic mappings \(f=h+\overline{g}\) from the class \(\mathcal {S}^0_H(\mathcal {S})\) which was recently introduced by Ponnusamy and Sairam Kaliraj. For example, we get the sharp estimate for \(|\arg h'(z)|\) in the case when \(|z| \leq 1/\sqrt{2}\) and obtain the sharp radius of convexity for \(h\). Our approach is applicable to a more general situation. Finally, we determine simple condition on the analytic part of univalent harmonic mappings so that it is in \(H_p\) spaces for \(0<p<1/3\).

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C55 General theory of univalent and multivalent functions of one complex variable
30H10 Hardy spaces
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abu-Muhanna, Y; Lyzzaik, A, The boundary behaviour of harmonic univalent maps, Pac. J. Math., 141, 1-20, (1990) · Zbl 0684.30018 · doi:10.2140/pjm.1990.141.1
[2] Aleman, A; Martin, MJ, Convex harmonic mappings are not necessarily in \(h^{1/2}\), Proc. Am. Math. Soc., 143, 755-763, (2015) · Zbl 1316.31001 · doi:10.1090/S0002-9939-2014-12281-7
[3] Avkhadiev, F.G., Wirths, K.-J.: Schwarz-Pick Type Inequalities, p. 156. Birkhäuser Verlag, Basel-Boston-Berlin (2009) · Zbl 1168.30001 · doi:10.1007/978-3-0346-0000-2
[4] Cima, JA; Livingston, AE, Integral smoothness properties of some harmonic mappings, Complex Var. Theory Appl., 11, 95-110, (1989) · Zbl 0724.30011
[5] Clunie, JG; Sheil-Small, T, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. AI, 9, 3-25, (1984) · Zbl 0506.30007
[6] Duren, P.L.: Theory of \(H^p\) Spaces, Pure and Applied Mathematics, vol. 38. Academic Press, New York and London (1970) · Zbl 0215.20203
[7] Duren, P.L.: Univalent Functions (Grundlehren der mathematischen Wissenschaften 259, Berlin, Heidelberg, Tokyo). Springer, New York (1983)
[8] Duren, P.L.: Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, vol. 156. Cambridge University Press, Cambridge (2004) · Zbl 1055.31001 · doi:10.1017/CBO9780511546600
[9] Feng, J; MacGregor, TH, Estimates of integral means of the derivatives of univalent functions, J. Anal. Math., 29, 203-231, (1976) · Zbl 0371.30014 · doi:10.1007/BF02789979
[10] Goluzin, GM, On distortion theorems in the theory of conformal mappings, Mat. Sb., 1, 127-135, (1936)
[11] Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs, vol. 26. American Mathematical Society, Providence (1969) · Zbl 0183.07502 · doi:10.1090/mmono/026
[12] Hedenmalm, H; Kayumov, IR, On the Makarov law of the iterated logarithm, Proc. Am. Math. Soc., 135, 2235-2248, (2007) · Zbl 1117.30024 · doi:10.1090/S0002-9939-07-08772-2
[13] Kayumov, IR, The law of the iterated logarithm for locally univalent functions, Ann. Acad. Sci. Fenn. Math., 27, 357-364, (2002) · Zbl 1078.30009
[14] Kayumov, IR, Lower estimates for integral means of univalent functions, Arkiv für Matematik, 44, 104-110, (2006) · Zbl 1163.30014 · doi:10.1007/s11512-005-0009-y
[15] Kayumov, IR, On brennan’s conjecture for a special class of functions, Math. Notes, 78, 498-502, (2005) · Zbl 1105.30009 · doi:10.1007/s11006-005-0149-1
[16] Makarov, NG, A note on the integral means of the derivative in conformal mapping, Proc. Am. Math. Soc., 96, 233-236, (1986) · Zbl 0623.30025 · doi:10.1090/S0002-9939-1986-0818450-0
[17] Nowak, M, Integral means of univalent harmonic maps, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 50, 155-162, (1996) · Zbl 0889.30019
[18] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992) · Zbl 0762.30001 · doi:10.1007/978-3-662-02770-7
[19] Ponnusamy, S., Rasila, A.: Planar Harmonic and Quasiregular Mappings, Topics in Modern Function Theory. In: Ruscheweyh, S., Ponnusamy, S. (eds.) Chapter in CMFT RMS-Lecture Notes Series, vol. 19, pp. 267-333 (2013) · Zbl 1318.30039
[20] Ponnusamy, S; Sairam Kaliraj, A, On the coefficient conjecture of clunie and sheil-small on univalent harmonic mappings, Proc. Indian Acad. Sci., 125, 277-290, (2015) · Zbl 1323.31002
[21] Ponnusamy, S; Yamamoto, H; Yanagihara, H, Variability regions for certain families of harmonic univalent mappings, Complex Var. Elliptic Equ., 58, 23-34, (2013) · Zbl 1294.30045 · doi:10.1080/17476933.2010.551200
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.