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Articles

Norm estimates of the partial derivatives and Schwarz lemma for α-harmonic functions

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Pages 1182-1194 | Received 16 Jan 2023, Accepted 17 Mar 2023, Published online: 26 Mar 2023

References

  • Garabedian PR. A partial differential equation arising in conformal mapping. Pac J Math. 1951;4:482–524.
  • Li P, Wang X, Xiao Q. Several properties of α-harmonic functions in the unit disk. Monatsh Math. 2017;184:627–640.
  • Li P, Rasila A, Wang Z. On properties of solutions to the α-harmonic equation. Complex Var Elliptic Equ. 2020;65(12):1981–1997.
  • Li M, Chen X. Schwarz lemma for solutions of the α-harmonic equation. Bull Malays Math Sci Soc. 2022;45:2691–2713.
  • Olofsson A, Wittsten J. Poisson integrals for standard weighted Laplacians in the unit disc. J Math Soc Jpn. 2013;65:447–486.
  • Olofsson A. Differential operators for a scale of Poisson type kernels in the unit disc. J Anal Math. 2014;123:227–249.
  • Duren P. Theory of Hp spaces. 2nd ed., Mineola (NY), Dover; 2000.
  • Duren P. Harmonic mappings in the plane. Cambridge: Cambridge Univ. Press; 2004.
  • Hedenmalm H, Korenblum B, Zhu K. Theory of bergman spaces. New York: Springer; 2000.
  • Zhu JF. Norm estimates of the partial derivatives for harmonic mappings and harmonic quasiregular mappings. J Geom Anal. 2021;31:5505–5525.
  • Chen SL, Ponnusamy S, Wang XT. Remarks on ‘Norm estimates of the partial derivatives for harmonic mappings and harmonic quasiregular mappings’. J Geom Anal. 2021;31:11051–11060.
  • Khalfallah A, Maleljević M. Estimates of partial derivatives for harmonic functions on the unit disc. Available at arxiv.org/abs/2302.09623.
  • Riesz M. Sur les fonctions conjugées. Math Zeit. 1927;27:218–244.
  • Rudin W. Real and complex analysis. Singapore: McGraw-Hill Book Co; 1966.
  • Hollenbeck B, Verbitsky IE. Best constants for the Riesz projection. J Fun Anal. 2000;175(2):370–392.
  • Kalaj D. On Riesz type inequalities for harmonic mappings on the unit disk. Trans Am Math Soc. 2019;372(6):4031–4051.
  • Khalfallah A, Mateljević M, Mhamdi M. Some properties of mappings admitting general Poisson representations. Mediterr J Math. 2021;18(5):1–19.
  • Heinz E. On one-to-one harmonic mappings. Pac J Math. 1959;9:101–105.
  • Hethcote HW. Schwarz lemma analogues for harmonic functions. Int J Math Ed Sci Tech. 1977;8(1):65–67.
  • Chen SL, Ponnusamy S. Schwarz's lemmas for mappings satisfying Poisson's equation. Indag Math (NS). 2019;30:1087–1098.
  • Khalfallah A, Haggui F, Mhamdi M. Generalized harmonic functions and Schwarz lemma for biharmonic mappings. Monatsh Math. 2021;196:823–849.
  • Khalfallah A, Mateljević M, Purtić B. Schwarz–Pick lemma for harmonic and hyperbolic harmonic functions. Results Math. 2022;77:167.
  • Li P, Li Y, Luo Q, et al. On Schwarz-Pick type inequality and lipschitz continuity for solutions to nonhomogeneous biharmonic equations. Mediterr J Math. 2023;20:142.
  • Li P, Luo Q, Ponnusamy S. Schwarz-Pick and landau type theorems for the solutions to Dirichlet-Neumann problem in the unit disk. Comput Methods Funct Theory. 2022;22:95–113.
  • Li P, Ponnusamy S. Representation formula and bi-Lipschitz continuity of solutions to inhomogeneous biharmonic dirichlet problems in the unit disk. J Math Anal Appl. 2017;456:1150–1175.
  • Mateljević M, Svetlik M. Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings. Appl Anal Discrete Math. 2020;14:150–168.

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