Some generalizations for the Schwarz-Pick lemma and boundary Schwarz lemma

Fangming Cai; Jie Rui; Deguang Zhong; Fangming Cai; Jie Rui; Deguang Zhong

doi:10.3934/math.20231586

AIMS Mathematics

2023, Volume 8, Issue 12: 30992-31007. doi: 10.3934/math.20231586

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Research article

Some generalizations for the Schwarz-Pick lemma and boundary Schwarz lemma

1.
School of Mathematics, Hunan University, Changsha 410082, China
2.
Department of Applied Statistics, Guangdong University of Finance, Guangzhou 510521, China
3.
Institute of Applied Mathematics, Shenzhen Polytechnic University, Shenzhen 518055, China

Received: 19 August 2023 Revised: 27 October 2023 Accepted: 31 October 2023 Published: 20 November 2023
MSC : 30C80, 31A30

In this paper, we first obtain a Schwarz-Pick type lemma for the holomorphic self-mapping of the unit disk with respect to the $ q $-distance. Second, we establish the general Schwarz-Pick lemma for the self-mapping of the unit disk satisfying the Poisson differential inequality. As an application, it is proven that this mapping is Lipschitz continuous with respect to the $ q $-distance under certain conditions. Moreover, the corresponding explicit Lipschitz constant is given. Third, it is proved that there exists a self-mapping of the unit disk satisfying the Poisson differential inequality, which does not meet conditions of the boundary Schwarz lemma. Finally, with some additional conditions, a boundary Schwarz lemma for the self-mapping of the unit disk satisfying the Poisson differential inequality is established.
- Schwarz-Pick lemma,
- Poisson differential inequality,
- $ q $-distance,
- boundary Schwarz lemma,
- explicit constant
Citation: Fangming Cai, Jie Rui, Deguang Zhong. Some generalizations for the Schwarz-Pick lemma and boundary Schwarz lemma[J]. AIMS Mathematics, 2023, 8(12): 30992-31007. doi: 10.3934/math.20231586

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Abstract

In this paper, we first obtain a Schwarz-Pick type lemma for the holomorphic self-mapping of the unit disk with respect to the $ q $-distance. Second, we establish the general Schwarz-Pick lemma for the self-mapping of the unit disk satisfying the Poisson differential inequality. As an application, it is proven that this mapping is Lipschitz continuous with respect to the $ q $-distance under certain conditions. Moreover, the corresponding explicit Lipschitz constant is given. Third, it is proved that there exists a self-mapping of the unit disk satisfying the Poisson differential inequality, which does not meet conditions of the boundary Schwarz lemma. Finally, with some additional conditions, a boundary Schwarz lemma for the self-mapping of the unit disk satisfying the Poisson differential inequality is established.

References

[1]	L. V. Ahlfors, An extension of Schwarz's lemma, T. Am. Math. Soc., 43 (1938), 359–364. https://doi.org/10.2307/1990065 doi: 10.2307/1990065
[2]	G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Monotonicity rules in calculus, Am. Math. Mon., 113 (2006), 805–816. https://doi.org/10.2307/27642062 doi: 10.2307/27642062
[3]	K. Astala, V. Manojlović, On Pavlović theorem in space, Potential Anal., 43 (2015), 361–370. https://doi.org/10.1007/s11118-015-9475-4 doi: 10.1007/s11118-015-9475-4
[4]	S. Bernstein, Sur la généralisation du problé de Dirichlet, Math. Ann., 62 (1906), 253–271. https://doi.org/10.1007/BF01449980 doi: 10.1007/BF01449980
[5]	M. Bonk, On Bloch's constant, P. Am. Math. Soc., 110 (1990), 889–894. https://doi.org/10.2307/2047734 doi: 10.2307/2047734
[6]	K. Broder, The Schwarz lemma: an odyssey, Rocky Mountain J. Math., 52 (2022), 1141–1155. https://doi.org/10.1216/rmj.2022.52.1141 doi: 10.1216/rmj.2022.52.1141
[7]	D. M. Burns, S. G. Krantz, Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Am. Math. Soc., 7 (1994), 661–676. https://doi.org/10.1090/S0894-0347-1994-1242454-2 doi: 10.1090/S0894-0347-1994-1242454-2
[8]	S. L. Chen, D. Kalaj, The Schwarz type lemmas and the Landau type theorem of mappings satisfying Poisson's equations, Complex Anal. Oper. Theory, 13 (2019), 2049–2068. https://doi.org/10.1007/s11785-019-00911-4 doi: 10.1007/s11785-019-00911-4
[9]	Z. H. Chen, Y. Liu, Y. F. Pan, A Schwarz lemma at the boundary of Hilbert balls, Chin. Ann. Math. Ser. B, 39 (2018), 695–704. https://doi.org/10.1007/s11401-018-0090-8 doi: 10.1007/s11401-018-0090-8
[10]	J. L. Chen, P. J. Li, S. K. Sahoo, X. T. Wang, On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains, Isr. J. Math., 220 (2017), 453–478. https://doi.org/10.1007/s11856-017-1522-y doi: 10.1007/s11856-017-1522-y
[11]	X. Chen, J. Wang, X. Tang, Boundary Schwarz lemma of the unit ball in $\mathbb{R}^{n}$ satisfying Poisson's equation, Acta Mathematica Sinica, Chinese Series, 66 (2023), 717–726. https://doi.org/10.12386/b20210679 doi: 10.12386/b20210679
[12]	P. Duren, Harmonic mappings in the plane, Cambridge: Cambridge University Press, 2004.
[13]	M. Elin, F. Jacobzon, M. Levenshtein, D. Shoikhet, The Schwarz lemma: rigidity and dynamics, In: Harmonic and complex analysis and its applications, Cham: Birkhäuser, 2014,135–230. https://doi.org/10.1007/978-3-319-01806-5_3
[14]	J. B. Garnett, Bounded analytic functions, New York: Academic Press, 1981.
[15]	I. Graham, H. Hamada, G. Kohr, A Schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings, J. Anal. Math., 140 (2020), 31–53. https://doi.org/10.1007/s11854-020-0080-0 doi: 10.1007/s11854-020-0080-0
[16]	H. Hamada, A Schwarz lemma at the boundary using the Julia-Wolff-Carathéodory type condition on finite dimensional irreducible bounded symmetric domains, J. Math. Anal. Appl., 465 (2018), 196–210. https://doi.org/10.1016/j.jmaa.2018.04.076 doi: 10.1016/j.jmaa.2018.04.076
[17]	H. Hamada, G. Kohr, A boundary Schwarz lemma for mappings from the unit polydisc to irreducible bounded symmetric domains, Math. Nachr., 293 (2020), 1345–1351. https://doi.org/10.1002/mana.201900493 doi: 10.1002/mana.201900493
[18]	E. Heinz, On certain nonlinear elliptic differential equations and univalent mappings, J. Anal. Math., 5 (1956), 197–272. https://doi.org/10.1007/BF02937346 doi: 10.1007/BF02937346
[19]	H. W. Hethcote, Schwarz lemma analogues for harmonic functions, Int. J. Math. Educ. Sci., 8 (1977), 65–67. https://doi.org/10.1080/0020739770080109 doi: 10.1080/0020739770080109
[20]	E. Hopf, Ueber den funktionalen, insbesondere den analytischen Charakter der Loesungen elliptischer Differentialgleichungen zweiter Ordnung, Math. Z., 34 (1932), 194–233. https://doi.org/10.1007/BF01180586 doi: 10.1007/BF01180586
[21]	D. Kalaj, A priori estimate of gradient of a solution to certain differential inequality and quasiconformal mappings, J. Anal. Math., 119 (2013), 63–88. https://doi.org/10.1007/s11854-013-0002-5 doi: 10.1007/s11854-013-0002-5
[22]	D. Kalaj, M. Mateljević, On certain nonlinear elliptic PDE and quasiconfomal mapps between euclidean surfaces, Potential Anal., 34 (2011), 13–22. https://doi.org/10.1007/s11118-010-9177-x doi: 10.1007/s11118-010-9177-x
[23]	D. Kalaj, On quasi-conformal harmonic maps between surfaces, Int. Math. Res. Notices, 2015 (2015), 355–380. https://doi.org/10.1093/imrn/rnt203 doi: 10.1093/imrn/rnt203
[24]	H. Li, M. Mateljević, Boundary Schwarz lemma for harmonic and pluriharmonic mappings in the unit ball, J. Math. Inequal., 16 (2022), 477–498. https://doi.org/10.7153/jmi-2022-16-35 doi: 10.7153/jmi-2022-16-35
[25]	T. Liu, X. Tang, A new boundary rigidity theorem for holomorphic self-mappings of the unit ball in $\mathbb{C}^{n}$, Pure Appl. Math. Q., 11 (2015), 115–130. https://doi.org/10.4310/PAMQ.2015.v11.n1.a5 doi: 10.4310/PAMQ.2015.v11.n1.a5
[26]	T. Liu, J. Wang, X. Tang, Schwarz lemma at the boundary of the unit ball in $\mathbb{C}^{n}$ and its applications, J. Geom. Anal., 25 (2015), 1890–1914. https://doi.org/10.1007/s12220-014-9497-y doi: 10.1007/s12220-014-9497-y
[27]	T. Liu, X. Tang, A boundary Schwarz lemma on the classical domain of type Ⅰ, Sci. China Math., 60 (2017), 1239–1258. https://doi.org/10.1007/s11425-015-0225-7 doi: 10.1007/s11425-015-0225-7
[28]	T. Liu, X. Tang, Schwarz lemma at the boundary of strongly pseudoconvex domain in $\mathbb{C}^{n}$, Math. Ann., 366 (2016), 655–666. https://doi.org/10.1007/s00208-015-1341-6 doi: 10.1007/s00208-015-1341-6
[29]	T. Liu, X. Tang, W. Zhang, Schwarz lemma at the boundary on the classical domain of type Ⅲ, Chin. Ann. Math. Ser. B, 41 (2020), 335–360. https://doi.org/10.1007/s11401-020-0202-0 doi: 10.1007/s11401-020-0202-0
[30]	M. Mateljević, Distortion of harmonic functions and harmonic quasiconformal quasi-isometry, Rev. Roumaine Math. Pures Appl., 51 (2006), 711–722.
[31]	M. Mateljević, M. Vuorinen, On harmonic quasiconformal quasi-isometries, J. Inequal. Appl., 2010 (2010), 178732. https://doi.org/10.1155/2010/178732 doi: 10.1155/2010/178732
[32]	M. Mateljević, Schwarz lemma and distortion for harmonic functions via length and area, Potential Anal., 53 (2020), 1165–1190. https://doi.org/10.1007/s11118-019-09802-x doi: 10.1007/s11118-019-09802-x
[33]	M. Mateljević, Boundary behaviour of partial derivatives for solutions to certain Laplacian-gradient inequalities and spatial QC maps, In: Operator theory and harmonic analysis, Cham: Springer, 2021,393–418. https://doi.org/10.1007/978-3-030-77493-6_23
[34]	M. Mateljević, A. Khalfallah, On some Schwarz type inequalities, J. Inequal. Appl., 2020 (2020), 164. https://doi.org/10.1186/s13660-020-02433-6 doi: 10.1186/s13660-020-02433-6
[35]	M. Mateljević, M. Svetlik, Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings, Appl. Anal. Discr. Math., 14 (2020), 150–168. https://doi.org/10.2298/AADM200104001M doi: 10.2298/AADM200104001M
[36]	M. Mateljević, N. Mutavdžić, The boundary Schwarz lemma for harmonic and pluriharmonic mappings and some generalizations, Bull. Malays. Math. Sci. Soc., 45 (2022), 3177–3195. https://doi.org/10.1007/s40840-022-01371-4 doi: 10.1007/s40840-022-01371-4
[37]	M. R. Mohapatra, X. T. Wang, J. F. Zhu, Boundary Schwarz lemma for solutions to nonhomogeneous biharmonic equations, B. Aust. Math. Soc., 100 (2019), 470–478. https://doi.org/10.1017/S0004972719000947 doi: 10.1017/S0004972719000947
[38]	M. R. Mohapatra, Schwarz-type lemma at the boundary for mappings satisfying non-homogeneous polyharmonic equations, Monatsh. Math., 192 (2020), 409–418. https://doi.org/10.1007/s00605-020-01402-x doi: 10.1007/s00605-020-01402-x
[39]	R. Osserman, A new variant of the Schwarz-Pick-Ahlfors lemma, Manuscripta Math., 100 (1999), 123–129. https://doi.org/10.1007/s002290050231 doi: 10.1007/s002290050231
[40]	G. Ren, Z. Xu, Schwarz's lemma for slice Clifford analysis, Adv. Appl. Clifford Algebras, 25 (2015), 965–976. https://doi.org/10.1007/s00006-015-0534-00 doi: 10.1007/s00006-015-0534-0
[41]	S. Ruscheweyh, Two remarks on bounded analytic functions, Serdica, 11 (1985), 200–202.
[42]	J. Schauder, Ueber lineate elliptische Differentialgleichungen zweiter Ordnung, Math. Z., 38 (1934), 257–282. https://doi.org/10.1007/BF01170635 doi: 10.1007/BF01170635
[43]	X. Tang, T. Liu, J. Zhang, Schwarz lemma at the boundary and rigidity property for holomorphic mappings on the unit ball of $\mathbb{C}^{n}$, P. Am. Math. Soc., 145 (2017), 1709–1716. https://doi.org/10.1090/proc/13378 doi: 10.1090/proc/13378
[44]	X. Tang, T. Liu, W. Zhang, Schwarz lemma at the boundary on the classical domain of type Ⅱ, J. Geom. Anal., 28 (2018), 1610–1634. https://doi.org/10.1007/s12220-017-9880-6 doi: 10.1007/s12220-017-9880-6
[45]	H. Werner, Das problem von Douglas fuer flaechen konstanter mittlerer Kruemmung, Math. Ann., 133 (1957), 303–319. https://doi.org/10.1007/BF01342884 doi: 10.1007/BF01342884
[46]	S. T. Yau, A general Schwarz lemma for Kähler manifolds, Am. J. Math., 100 (1978), 197–203. https://doi.org/10.2307/2373880 doi: 10.2307/2373880
[47]	D. G. Zhong, F. N. Meng, W. J. Yuan, On Schwarz-Pick inequality mappings satisfying Poisson differential inequality, Acta Math. Sci., 41 (2021), 959–967. https://doi.org/10.1007/s10473-021-0320-0 doi: 10.1007/s10473-021-0320-0
[48]	D. G. Zhong, M. L. Huang, D. P. Wei, Some inequalities for self-mappings of unit ball satisfying the invariant Laplacians, in press.
[49]	J. F. Zhu, Schwarz lemma and boundary Schwarz lemma for pluriharmonic mappings, Filomat, 32 (2018), 5385–5402. https://doi.org/10.2298/FIL1815385Z doi: 10.2298/FIL1815385Z
[50]	J. F. Zhu, X. T. Wang, Boundary Schwarz lemma for solutions to Poisson's equation, J. Math. Anal. Appl., 463 (2018), 623–633. https://doi.org/10.1016/j.jmaa.2018.03.043 doi: 10.1016/j.jmaa.2018.03.043

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