PRECISE VALUES OF THE BLOCH CONSTANTS OF CERTAIN LOG-p-HARMONIC MAPPINGS

doi:10.1007/s10473-021-0117-1

Abstract

Abstract: The aim of this article is twofold. One aim is to establish the precise forms of Landau-Bloch type theorems for certain polyharmonic mappings in the unit disk by applying a geometric method. The other is to obtain the precise values of Bloch constants for certain log-p-harmonic mappings. These results improve upon the corresponding results given in Bai et al. (Complex Anal. Oper. Theory, 13(2): 321-340, 2019).

Key words: Bloch constant, Landau theorem, Bloch theorem, biharmonic mappings, polyharmonic mappings, log-p-harmonic mappings, univalent

CLC Number:

30C99

Mingsheng LIU, Lifang LUO. PRECISE VALUES OF THE BLOCH CONSTANTS OF CERTAIN LOG-p-HARMONIC MAPPINGS[J].Acta mathematica scientia,Series B, 2021, 41(1): 297-310.

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References

[1] Abdulhadi Z, Muhanna Y, Khuri S. On univalent solutions of the biharmonic equations. J Inequal Appl, 2005, 5:469-478
[2] Abdulhadi Z, Muhanna Y. Landau's theorems for biharmonic mappings. J Math Anal Appl, 2008, 338:705-709
[3] Bai X X, Liu M S. Landau-type theorems of polyharmonic mappings and log-p-harmonic mappings. Complex Anal Oper Theory, 2019, 13(2):321-340
[4] Chen H-H, Gauthier P M, Hengartner W. Bloch constants for planar harmonic mappings. Proc Amer Math Soc, 2000, 128(11):3231-3240
[5] Chen H-H, Gauthier P. The Landau theorem and Bloch theorem for planar Harmonic and pluriharmonic mappings. Proc Amer Math Soc, 2011, 139:583-595
[6] Chen S H, Ponnusamy S, Wang X. Landau's theorems for certain biharmonic mappings. Appl Math Comput, 2009, 208(2):427-433
[7] Chen S H, Ponnusamy S, Wang X. Bloch constant and Landau's theorem for planar p-harmonic mappings. J Math Appl Appl, 2011, 373:102-110
[8] Chen S H, Ponnusamy S, Wang X. Properties of some classes of planar harmonic and planar biharmonic mappings. Complex Anal Oper Theory, 2011, 5:901-916
[9] Chen S H, Ponnusamy S, Rasila A. Coefficient estimates, Landau's theorem and Lipschitz-type spaces on planar harmonic mappings. J Aust Math Soc, 2014, 96(2):198-215
[10] Chen S H, Ponnusamy S, Wang X. Coefficient estimates and Landau-Bloch's constant for planar harmonic mappings. Bull Malaysian Math Sci Soc Second Series, 2011, 34(2):255-265
[11] Chen J, Rasila A, Wang X. Landau's theorem for polyharmonic mappings. J Math Appl Appl, 2014, 409:934-945
[12] Clunie J G, Sheil-Small T. Harmonic univalent functions. Ann Acad Sci Fenn, Ser AI, 1984, 9:3-25
[13] Dorff M, Nowak M. Landau's theorem for planar harmonic mappings. Comput Meth Funct Theory, 2004, 4(1):151-158
[14] Graham I, Kohr G. Geometric Function Theory in One and Higher Dimensions. New York:Marcel Dekker Inc, 2003
[15] Grigoryan A. Landau and Bloch theorems for harmonic mappings. Complex Variable Theory Appl, 2006, 51(1):81-87
[16] He L, Tu Z H. The schwarz lemma at the boundary of the non-convex complex ellipsoids. Acta Math Sci, 2019, 39B(4):915-926
[17] Huang X Z. Estimates on Bloch constants for planar harmonic mappings. J Math Anal Appl, 2008, 337:880-887.
[18] Huang X Z. Sharp estimate on univalent radius for planar harmonic mappings with bounded Fréchet derivative (in Chinese). Sci Sin Math, 2014, 446:685-692
[19] Lewy H. On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull Amer Math Soc, 1936, 42:689-692
[20] Li P, Ponnusamy S, Wang X. Some properties of planar p-harmonic and log-p-harmonic mappings. Bull Malays Math Sci Soc, 2013, 36(3):595-609
[21] Li P, Wang X. Landau's theorem for log-p-harmonic mappings. Appl Math Comput, 2012, 218:4806-4812
[22] Liu M S. Landau's theorems for biharmonic mappings. Complex Variables and Elliptic Equations, 2008, 53(9):843-855
[23] Liu M S. Estimates on Bloch constants for planar harmonic mappings. Sci China Ser A-Math, 2009, 52(1):87-93
[24] Liu M S. Landau's theorem for planar harmonic mappings. Comput Math Appl, 2009, 57(7):1142-1146
[25] Liu M S, Chen H-H. The Landau-Bloch type theorems for planar harmonic mappings with bounded dilation. J Math Anal Appl, 2018, 468(2):1066-1081
[26] Liu M S, Luo L F, Luo X. Landau-Bloch type theorems for strongly bounded harmonic mappings. Monatshefte für Mathematik, 2020, 191(1):175-185
[27] Liu M S, Liu Z X. Landau-type theorems for p-harmonic mappings or log-p-harmonic mappings. Applied Analysis, 2014, 51(1):81-87
[28] Liu M S, Wu F, Yang Y. Sharp estimates of quasi-convex mappings of type B and order alpha. Acta Math Sci, 2019, 39 B(5):1265-1276
[29] Liu M S, Xie L, Yang L M. Landau's theorems for biharmonic mappings (Ⅱ). Mathematical Methods Applied Sciences, 2017, 40:2582-2595
[30] Liu M S, Luo L F. Landau-type theorems for certain bounded biharmonic mappings. Results in Mathematics, 2019, 74, Art 170
[31] Xia X Q, Huang X Z. Estimates on Bloch constants for planar bounded harmonic mappings (in Chinese). Chinese Annals of Mathematics A, 2010, 31(6):769-776
[32] Zhu J F. Landau theorem for planar harmonic mappings. Complex Anal Oper Theory, 2015, 9:1819-1826
[33] Zhu Y C, Liu M S. Landau-type theorems for certain planar harmonic mappings or biharmonic mappings. Complex Variables and Elliptic Equations, 2013, 58(12):1667-1676