Properties for ($\alpha,\beta$)-harmonic functions

Antti Rasila; Jiale Chang; Jinjing Qiao

arxiv: 2512.04379 · v2 · submitted 2025-12-04 · 🧮 math.CV

Properties for (α,β)-harmonic functions

Jinjing Qiao , Jiale Chang , Antti Rasila This is my paper

Pith reviewed 2026-05-17 02:03 UTC · model grok-4.3

classification 🧮 math.CV

keywords (α,β)-harmonic functionsHeinz inequalitycoefficient boundsstarlike mappingsRadó theoremKoebe theoremarea theoremharmonic mappings

0 comments

The pith

(α,β)-harmonic functions satisfy Heinz's inequality and inherit Radó's theorem plus Koebe covering results from ordinary harmonic functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes analytic properties for (α,β)-harmonic functions by first deriving Heinz's inequality and a coefficient bound for normalized univalent members of the class, with the bound holding in the starlike subclass. It then uses a direct relationship to standard harmonic functions to recover Radó's theorem, Koebe-type covering theorems, and an area theorem. Growth and distortion estimates are additionally obtained from L^p norms of the boundary functions. A sympathetic reader would care because these steps extend classical results in geometric function theory to a parameterized family without needing new proofs for each parameter choice.

Core claim

We obtain Heinz's inequality for (α,β)-harmonic functions, propose a coefficient bound for normalized univalent (α,β)-harmonic functions and prove that this holds for the subclass that consists of starlike functions. Furthermore, by utilizing the relationship between (α,β)-harmonic functions and harmonic functions, we obtain Radó's theorem, Koebe type covering theorems and an area theorem. Finally, we show growth estimates and distortion estimates for (α,β)-harmonic functions by using the L^p norms of the boundary functions.

What carries the argument

The relationship between (α,β)-harmonic functions and ordinary harmonic functions that permits transferring theorems like Radó's and Koebe-type results without parameter-specific adjustments.

If this is right

Heinz's inequality holds for (α,β)-harmonic functions.
A coefficient bound is valid for normalized univalent (α,β)-harmonic functions and remains valid inside the starlike subclass.
Radó's theorem applies to (α,β)-harmonic functions.
Koebe-type covering theorems and the area theorem extend to the class through the relationship.
Growth and distortion estimates follow from the L^p norms of the boundary functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Varying α and β may generate families whose distortion or covering radii can be tuned for use in approximation or mapping problems.
If the relationship is uniform in the parameters, other classical results from univalent function theory could transfer in the same way.
The class could provide controlled examples for testing conjectures on harmonic mappings when α and β are chosen to alter the analytic form.

Load-bearing premise

The relationship between (α,β)-harmonic functions and ordinary harmonic functions is sufficiently strong and direct to transfer Radó's theorem, Koebe-type results, and the area theorem without additional restrictions or counterexamples arising from the parameters α and β.

What would settle it

A concrete (α,β)-harmonic function for distinct α and β that violates the Koebe covering radius or the area theorem would disprove the direct transfer of those results.

read the original abstract

We investigate properties of ($\alpha,\beta$)-harmonic functions. First, we discuss the coefficient estimates for ($\alpha,\beta$)-harmonic functions. In particular, we obtain Heinz's inequality for ($\alpha,\beta$)-harmonic functions, propose a coefficient bound for normalized univalent ($\alpha,\beta$)-harmonic functions and prove that this holds for the subclass that consists of starlike functions. Furthermore, by utilizing the relationship between ($\alpha,\beta$)-harmonic functions and harmonic functions, we obtain Rad\'{o}'s theorem, Koebe type covering theorems and an area theorem. Finally, we show growth estimates and distortion estimates for ($\alpha,\beta$)-harmonic functions by using the $L^p$ norms of the boundary functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper pulls over Heinz inequality, a coefficient bound for starlike cases, and transfers of Radó, Koebe, and area theorems to the (α,β)-harmonic class using its link to ordinary harmonic functions, but the transfers rest on an unexamined parameter dependence.

read the letter

The core contribution is straightforward: they introduce coefficient estimates for (α,β)-harmonic functions, prove Heinz's inequality in this setting, give a bound that holds for the normalized univalent ones and specifically for the starlike subclass, then invoke the relationship to standard harmonic mappings to carry over Radó's theorem, Koebe-type covering results, and an area theorem. They close with growth and distortion estimates from L^p norms on the boundary functions. That is the actual new material on the page, and it is presented cleanly enough for readers already working in this corner of geometric function theory to follow the steps without extra machinery. The estimates themselves look like routine but competent applications of the usual techniques once the class is defined. Credit for shipping explicit statements for the parameterized case rather than just gesturing at it. The soft spot is exactly where the stress-test flagged it. The transfers of the classical theorems depend on the (α,β)-to-harmonic correspondence preserving univalence, starlikeness, and the relevant boundary behavior. If that correspondence is not bijective or property-preserving for every admissible α and β, then the transferred results need extra hypotheses or counterexamples. The abstract gives no sign that the authors checked the edges or stated the precise range, so a referee would reasonably ask for that clarification in the full text. Nothing else looks broken. This is a specialized note for people already reading papers on harmonic mappings and their generalizations. It is not going to reorganize the field, but the concrete bounds and transferred theorems are the sort of incremental results that keep the literature moving. A serious editor should send it to peer review rather than desk-reject; the claims are specific enough that referees can check the derivations and the parameter handling in one pass.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates properties of (α,β)-harmonic functions in the unit disk. It establishes coefficient estimates, including Heinz's inequality for this class, and proposes a coefficient bound for normalized univalent (α,β)-harmonic functions that is shown to hold for the starlike subclass. By invoking a relationship between (α,β)-harmonic functions and ordinary harmonic functions, the authors derive Radó's theorem, Koebe-type covering theorems, and an area theorem. The paper concludes with growth and distortion estimates obtained via L^p norms of the boundary functions.

Significance. If the central relationship to standard harmonic functions is shown to preserve the relevant properties uniformly, the work would extend several classical results from harmonic mapping theory to a parameterized family. The coefficient estimates and the transfer of Radó, Koebe, and area theorems could provide a useful framework for studying generalized harmonic mappings, particularly if the derivations are parameter-explicit and the bounds are sharp.

major comments (1)

[Section on relationship to harmonic functions (post-coefficient estimates)] The derivation of Radó's theorem, Koebe-type covering theorems, and the area theorem rests entirely on the relationship between (α,β)-harmonic functions and ordinary harmonic functions (discussed after the coefficient estimates). The manuscript does not state the admissible range of α and β, nor does it verify that the correspondence preserves univalence, starlikeness, or the necessary boundary regularity for all such parameters. If the mapping depends on α and β in a way that alters these properties outside a restricted interval, the transferred theorems require additional hypotheses or counterexamples.

minor comments (2)

[Introduction / Definition section] The definition of (α,β)-harmonicity and the precise normalization condition for the univalent subclass should be stated explicitly at the outset, including any restrictions on α and β, to make the subsequent claims self-contained.
[Final section on growth and distortion estimates] Notation for the L^p norms of the boundary functions in the growth and distortion estimates could be clarified, particularly how these norms relate back to the (α,β) parameters.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive major comment. We address the point raised below and outline the revisions we will make.

read point-by-point responses

Referee: [Section on relationship to harmonic functions (post-coefficient estimates)] The derivation of Radó's theorem, Koebe-type covering theorems, and the area theorem rests entirely on the relationship between (α,β)-harmonic functions and ordinary harmonic functions (discussed after the coefficient estimates). The manuscript does not state the admissible range of α and β, nor does it verify that the correspondence preserves univalence, starlikeness, or the necessary boundary regularity for all such parameters. If the mapping depends on α and β in a way that alters these properties outside a restricted interval, the transferred theorems require additional hypotheses or counterexamples.

Authors: We agree with the referee that the admissible range of the parameters α and β must be stated explicitly and that the preservation of univalence, starlikeness, and boundary regularity under the correspondence to ordinary harmonic functions requires verification. In the revised manuscript we will insert a short preliminary subsection immediately before the derivations of Radó’s theorem, the Koebe-type covering theorems, and the area theorem. There we will specify that α, β ∈ ℝ satisfy |α| + |β| < 1 (the range already implicit in the definition of (α,β)-harmonic mappings used throughout the paper) and supply a brief argument showing that, under this restriction, the correspondence preserves sense-preservation, univalence, starlikeness, and the requisite boundary regularity. With these additions the transferred theorems will hold rigorously for the class under consideration. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results transferred via external relationship to standard harmonic functions

full rationale

The paper defines (α,β)-harmonic functions and obtains coefficient estimates, Heinz inequality, and bounds for univalent/starlike subclasses directly. It then transfers Radó's theorem, Koebe-type covering theorems, and the area theorem by invoking a relationship to ordinary harmonic functions, followed by growth/distortion estimates via L^p boundary norms. This relationship functions as an independent mapping tool rather than a self-referential construction. No quoted steps reduce a claimed result to its own inputs by definition, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation remains self-contained against external results in harmonic function theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of a direct functional relationship allowing transfer of harmonic-function theorems, plus standard analytic properties of univalent and starlike mappings. No new entities are postulated. α and β function as fixed parameters of the class rather than fitted values.

axioms (1)

domain assumption There exists a sufficiently direct relationship between (α,β)-harmonic functions and ordinary harmonic functions that permits direct transfer of Radó's theorem, Koebe covering theorems, and the area theorem.
Invoked explicitly to obtain the listed classical results for the generalized class.

pith-pipeline@v0.9.0 · 5420 in / 1407 out tokens · 34499 ms · 2026-05-17T02:03:13.367151+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We focus on the following associated homogeneous equation on D: L_{α,β}u=0. ... u is (α,β)-harmonic if it satisfies (1.1).
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By utilizing the relationship between (α,β)-harmonic functions and harmonic functions, we obtain Radó’s theorem, Koebe type covering theorems and an area theorem.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

Ahern, J

P. Ahern, J. Bruna, C. Cascante.H p-theory for generalizedM-harmonic functions in the unit ball. Indiana Univ. Math. J. (1996), 45(1), 103-135

work page 1996
[2]

Andrews, R

G. Andrews, R. Askey, R. Roy. Special functions. Encyclopedia Math. Appl., Cambridge Uni- versity Press, 1999

work page 1999
[3]

Arsenovi´ c, J

M. Arsenovi´ c, J. Gaji´ c, Schwarz-Pick lemma for (α, β)-harmonic functions in the unit disc. J. Math. Anal. Appl. (2024), 539(1), Article No. 128489

work page 2024
[4]

Bshouty, S

D. Bshouty, S. Evdoridis, A. Rasila. A note on Polyharmonic mappings. Comput. Methods Funct. Theory (2021), 22(3), 1–11

work page 2021
[5]

Carlsson, J

M. Carlsson, J. Wittsten. The Dirichlet problem for standard weighted Laplacians in the upper half plane. J. Math. Anal. Appl. (2016), 436, 868–889

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J. Chen, S. Chen, M. Huang, H. Zheng. Isoperimetric type inequalities for mappings induced by weighted Laplace differential operators. J. Geom. Anal. (2023), 33(7), Article No. 216

work page 2023
[7]

S. Chen, S. Ponnusamy, X. Wang. Covering and distortion theorems for planar harmonic univalent mappings. Arch. Math. (Basel) (2013), 101(3), 285–291

work page 2013
[8]

J. G. Clunie, T. Sheil-Small. Harmonic univalent functions. Ann. Acad. Sci. Fenn. Math. (1984), 9, 3–25

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P. Duren. Theory ofH p spaces. Academic Press, New York and London, 1970

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P. Duren. Univalent functions. Spring-Verlag, New York, 1983

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P. Duren. Harmonic mappings in the plane. Cambridge University, Cambridge, 2004

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[12]

D. Geller. Some results inH p theory for the Heisenberg group. Duke Math. J. (1980), 47(2), 365–390

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[13]

Heikkala, M

V. Heikkala, M. K. Vamanamurthy, M. Vuorinen. Generalized elliptic integrals. Comput. Meth. Funct. Theory (2009), 9(1), 75–109

work page 2009
[14]

D. Kalaj. Sharp pointwise estimate ofα-harmonic functions. Complex Anal. Oper. Theory (2024), 18(6), No. 135

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[15]

Kalaj, M

D. Kalaj, M. Markovi´ c. Optimal estimates for the gradient of harmonic functions in the unit disk. Complex Anal. (2013), 7(4), 1167–1183

work page 2013
[16]

Khalfallah, M

A. Khalfallah, M. Mhamdi. Estimates of the first partial derivatives of (α, β)-harmonic func- tions on the unit disc. J. Inequal. Appl. (2025), 2025, Article No.109

work page 2025
[17]

Khalfallah, M

A. Khalfallah, M. Mateljevi¡¨ ac. Norm estimates of the partial derivatives and Schwarz lemma forα-harmonic functions. Complex Var. Elliptic Equ. (2024), 69(7), 1182–1194

work page 2024
[18]

Khalfallah, M

A. Khalfallah, M. Mateljevi´ c. Estimates of partial derivatives for harmonic functions on the unit disc. Comput. Methods Funct. Theory (2024), 24(4), 883–893

work page 2024
[19]

Kneser, L¨ osung der Aufgabe 41

H. Kneser, L¨ osung der Aufgabe 41. Jahresber. Deutsch. Math. Verein. (1926), 35, 123–124

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[20]

Klintborg, A

M. Klintborg, A. Olofsson. A series expansion for generalized harmonic functions. Anal. Math. Phys. (2021), 11(2), Article No. 122

work page 2021
[21]

R. R. Hall, On an inequality of E. Heinz. J. Analyse Math. (1982/83), 42, 185–198

work page 1982
[22]

P. Li, X. Wang, Q. Xiao. Several properties ofα-harmonic functions in the unit disk. Monatsh. Math. (2017), 184, 627–640

work page 2017
[23]

P. Li, A. Rasila, Z. Wang. On properties of solutions to theα-harmonic equation. Complex Var. Elliptic Equ. (2020), 65(12), 1981–1997

work page 2020
[24]

P. Li, Q. Luo, S. Ponnusamy. On properties of the solutions to the (p, q)-harmonic functions. J. Math. Anal. Appl. (2025), 549(1), Artile No. 129437

work page 2025
[25]

B. Long, Q. Wang. Some coefficient estimates on real kernelα-harmonic mappings. Proc. Amer. Math. Soc. (2022), 150(4), 1529–1540

work page 2022
[26]

B. Long. Boundary behavior ofα-harmonic functions and their Riesz-Fej´ er inequalities. arXiv: 2410.12137 (2024)

work page arXiv 2024
[27]

B. Long. Some optimal inequalities forα-harmonic functions estimated by their boundary functions. Potential Anal. (2025). Doi.org/10.1007/s11118-025-10221-4

work page doi:10.1007/s11118-025-10221-4 2025
[28]

Abu Muhanna, R

Y. Abu Muhanna, R. M. Ali, S. Ponnusamy. The spherical metric and univalent harmonic mappings. Monatsh. Math. (2019), 188, 703–716. (α, β)-harmonic functions 33

work page 2019
[29]

Olofsson

A. Olofsson. Differential operators for a scale of Poisson type kernels in the unit disc. J. Anal. Math. (2014), 123, 227–249

work page 2014
[30]

Olofsson, J

A. Olofsson, J. Wittsten. Poisson integrals for standard weighted Laplacians in the unit disc. J. Math. Soc. Japan (2013), 65, 447–486

work page 2013
[31]

Rudin, Real and complex analysis (Third Edition)

W. Rudin, Real and complex analysis (Third Edition). McGraw-Hill Book Company, New York, 1987

work page 1987
[32]

Sheil-Small, Constants for planar harmonic mappings

T. Sheil-Small, Constants for planar harmonic mappings. J. London Math. Soc. (1990), 42, 237–248

work page 1990
[33]

Zh. Wang, X. Wang, A. Rasila, J. Qiu, On a problem of Pavlovi´ c involving harmonic quasi- conformal mappings. arXiv:2405.19852 (2024). J. Qiao, Department of Mathematics, College of Mathematics and Information Science, Hebei University, Baoding 071002, Hebei, People’s Republic of China. Hebei Key Laboratory of Machine Learning and Computational Intellige...

work page arXiv 2024

[1] [1]

Ahern, J

P. Ahern, J. Bruna, C. Cascante.H p-theory for generalizedM-harmonic functions in the unit ball. Indiana Univ. Math. J. (1996), 45(1), 103-135

work page 1996

[2] [2]

Andrews, R

G. Andrews, R. Askey, R. Roy. Special functions. Encyclopedia Math. Appl., Cambridge Uni- versity Press, 1999

work page 1999

[3] [3]

Arsenovi´ c, J

M. Arsenovi´ c, J. Gaji´ c, Schwarz-Pick lemma for (α, β)-harmonic functions in the unit disc. J. Math. Anal. Appl. (2024), 539(1), Article No. 128489

work page 2024

[4] [4]

Bshouty, S

D. Bshouty, S. Evdoridis, A. Rasila. A note on Polyharmonic mappings. Comput. Methods Funct. Theory (2021), 22(3), 1–11

work page 2021

[5] [5]

Carlsson, J

M. Carlsson, J. Wittsten. The Dirichlet problem for standard weighted Laplacians in the upper half plane. J. Math. Anal. Appl. (2016), 436, 868–889

work page 2016

[6] [6]

J. Chen, S. Chen, M. Huang, H. Zheng. Isoperimetric type inequalities for mappings induced by weighted Laplace differential operators. J. Geom. Anal. (2023), 33(7), Article No. 216

work page 2023

[7] [7]

S. Chen, S. Ponnusamy, X. Wang. Covering and distortion theorems for planar harmonic univalent mappings. Arch. Math. (Basel) (2013), 101(3), 285–291

work page 2013

[8] [8]

J. G. Clunie, T. Sheil-Small. Harmonic univalent functions. Ann. Acad. Sci. Fenn. Math. (1984), 9, 3–25

work page 1984

[9] [9]

P. Duren. Theory ofH p spaces. Academic Press, New York and London, 1970

work page 1970

[10] [10]

P. Duren. Univalent functions. Spring-Verlag, New York, 1983

work page 1983

[11] [11]

P. Duren. Harmonic mappings in the plane. Cambridge University, Cambridge, 2004

work page 2004

[12] [12]

D. Geller. Some results inH p theory for the Heisenberg group. Duke Math. J. (1980), 47(2), 365–390

work page 1980

[13] [13]

Heikkala, M

V. Heikkala, M. K. Vamanamurthy, M. Vuorinen. Generalized elliptic integrals. Comput. Meth. Funct. Theory (2009), 9(1), 75–109

work page 2009

[14] [14]

D. Kalaj. Sharp pointwise estimate ofα-harmonic functions. Complex Anal. Oper. Theory (2024), 18(6), No. 135

work page 2024

[15] [15]

Kalaj, M

D. Kalaj, M. Markovi´ c. Optimal estimates for the gradient of harmonic functions in the unit disk. Complex Anal. (2013), 7(4), 1167–1183

work page 2013

[16] [16]

Khalfallah, M

A. Khalfallah, M. Mhamdi. Estimates of the first partial derivatives of (α, β)-harmonic func- tions on the unit disc. J. Inequal. Appl. (2025), 2025, Article No.109

work page 2025

[17] [17]

Khalfallah, M

A. Khalfallah, M. Mateljevi¡¨ ac. Norm estimates of the partial derivatives and Schwarz lemma forα-harmonic functions. Complex Var. Elliptic Equ. (2024), 69(7), 1182–1194

work page 2024

[18] [18]

Khalfallah, M

A. Khalfallah, M. Mateljevi´ c. Estimates of partial derivatives for harmonic functions on the unit disc. Comput. Methods Funct. Theory (2024), 24(4), 883–893

work page 2024

[19] [19]

Kneser, L¨ osung der Aufgabe 41

H. Kneser, L¨ osung der Aufgabe 41. Jahresber. Deutsch. Math. Verein. (1926), 35, 123–124

work page 1926

[20] [20]

Klintborg, A

M. Klintborg, A. Olofsson. A series expansion for generalized harmonic functions. Anal. Math. Phys. (2021), 11(2), Article No. 122

work page 2021

[21] [21]

R. R. Hall, On an inequality of E. Heinz. J. Analyse Math. (1982/83), 42, 185–198

work page 1982

[22] [22]

P. Li, X. Wang, Q. Xiao. Several properties ofα-harmonic functions in the unit disk. Monatsh. Math. (2017), 184, 627–640

work page 2017

[23] [23]

P. Li, A. Rasila, Z. Wang. On properties of solutions to theα-harmonic equation. Complex Var. Elliptic Equ. (2020), 65(12), 1981–1997

work page 2020

[24] [24]

P. Li, Q. Luo, S. Ponnusamy. On properties of the solutions to the (p, q)-harmonic functions. J. Math. Anal. Appl. (2025), 549(1), Artile No. 129437

work page 2025

[25] [25]

B. Long, Q. Wang. Some coefficient estimates on real kernelα-harmonic mappings. Proc. Amer. Math. Soc. (2022), 150(4), 1529–1540

work page 2022

[26] [26]

B. Long. Boundary behavior ofα-harmonic functions and their Riesz-Fej´ er inequalities. arXiv: 2410.12137 (2024)

work page arXiv 2024

[27] [27]

B. Long. Some optimal inequalities forα-harmonic functions estimated by their boundary functions. Potential Anal. (2025). Doi.org/10.1007/s11118-025-10221-4

work page doi:10.1007/s11118-025-10221-4 2025

[28] [28]

Abu Muhanna, R

Y. Abu Muhanna, R. M. Ali, S. Ponnusamy. The spherical metric and univalent harmonic mappings. Monatsh. Math. (2019), 188, 703–716. (α, β)-harmonic functions 33

work page 2019

[29] [29]

Olofsson

A. Olofsson. Differential operators for a scale of Poisson type kernels in the unit disc. J. Anal. Math. (2014), 123, 227–249

work page 2014

[30] [30]

Olofsson, J

A. Olofsson, J. Wittsten. Poisson integrals for standard weighted Laplacians in the unit disc. J. Math. Soc. Japan (2013), 65, 447–486

work page 2013

[31] [31]

Rudin, Real and complex analysis (Third Edition)

W. Rudin, Real and complex analysis (Third Edition). McGraw-Hill Book Company, New York, 1987

work page 1987

[32] [32]

Sheil-Small, Constants for planar harmonic mappings

T. Sheil-Small, Constants for planar harmonic mappings. J. London Math. Soc. (1990), 42, 237–248

work page 1990

[33] [33]

Zh. Wang, X. Wang, A. Rasila, J. Qiu, On a problem of Pavlovi´ c involving harmonic quasi- conformal mappings. arXiv:2405.19852 (2024). J. Qiao, Department of Mathematics, College of Mathematics and Information Science, Hebei University, Baoding 071002, Hebei, People’s Republic of China. Hebei Key Laboratory of Machine Learning and Computational Intellige...

work page arXiv 2024