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Journal articles on the topic 'Schwarz Lemma and generalization'

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Published: 10 March 2023

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1

Joseph, James E., and Myung H. Kwack. "A Generalization of the Schwarz Lemma to Normal Selfaps of Complex Spaces." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 68, no. 1 (February 2000): 10–18. http://dx.doi.org/10.1017/s1446788700001543.

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2

Svetlik, Marek. "A note on the Schwarz lemma for harmonic functions." Filomat 34, no. 11 (2020): 3711–20. http://dx.doi.org/10.2298/fil2011711s.

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In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point z = 0 are given.

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3

Roth, Oliver. "The Nehari-Schwarz lemma and infinitesimal boundary rigidity of bounded holomorphic functions." Studia Universitatis Babes-Bolyai Matematica 67, no. 2 (June 8, 2022): 285–94. http://dx.doi.org/10.24193/subbmath.2022.2.05.

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"We survey a number of recent generalizations and sharpenings of Nehari's extension of Schwarz' lemma for holomorphic self{maps of the unit disk. In particular, we discuss the case of in nitely many critical points and its relation to the zero sets and invariant subspaces for Bergman spaces, as well as the case of equality at the boundary."

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4

Bisi, Cinzia, and Caterina Stoppato. "Landau’s theorem for slice regular functions on the quaternionic unit ball." International Journal of Mathematics 28, no. 03 (March 2017): 1750017. http://dx.doi.org/10.1142/s0129167x17500173.

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During the development of the theory of slice regular functions over the real algebra of quaternions [Formula: see text] in the last decade, some natural questions arose about slice regular functions on the open unit ball [Formula: see text] in [Formula: see text]. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of [Formula: see text] fixing the origin, it establishes two variants of the quaternionic Schwarz–Pick lemma, specialized to maps [Formula: see text] that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps [Formula: see text] of the complex unit disk with [Formula: see text]. Landau had computed, in terms of [Formula: see text], a radius [Formula: see text] such that [Formula: see text] is injective at least in the disk [Formula: see text] and such that the inclusion [Formula: see text] holds. The analogous result proven here for slice regular functions [Formula: see text] allows a new approach to the study of Bloch–Landau-type properties of slice regular functions [Formula: see text].

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5

Zhu, Jian-Feng. "Schwarz lemma and boundary Schwarz lemma for pluriharmonic mappings." Filomat 32, no. 15 (2018): 5385–402. http://dx.doi.org/10.2298/fil1815385z.

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In this paper, we first improve the boundary Schwarz lemma for holomorphic self-mappings of the unit ball Bn, and then we establish the boundary Schwarz lemma for harmonic self-mappings of the unit disk D and pluriharmonic self-mappings of Bn. The results are sharp and coincides with the classical boundary Schwarz lemma when n = 1.

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6

Yang, Yan, and Tao Qian. "Schwarz lemma in Euclidean spaces." Complex Variables and Elliptic Equations 51, no. 7 (July 2006): 653–59. http://dx.doi.org/10.1080/17476930600688623.

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7

Edigarian, Armen, and Włodzimierz Zwonek. "Schwarz lemma for the tetrablock." Bulletin of the London Mathematical Society 41, no. 3 (March 22, 2009): 506–14. http://dx.doi.org/10.1112/blms/bdp022.

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8

Ratto, Andrea, Marco Rigoli, and Laurent Veron. "extensions of the Schwarz Lemma." Duke Mathematical Journal 74, no. 1 (April 1994): 223–36. http://dx.doi.org/10.1215/s0012-7094-94-07411-5.

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9

Xu, Zhenghua. "Schwarz lemma for pluriharmonic functions." Indagationes Mathematicae 27, no. 4 (September 2016): 923–29. http://dx.doi.org/10.1016/j.indag.2016.06.002.

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10

Huang, Ziyan, Di Zhao, and Hongyi Li. "A boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions." Filomat 34, no. 9 (2020): 3151–60. http://dx.doi.org/10.2298/fil2009151h.

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In this paper, we present a boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions, which extends the classical Schwarz lemma for bounded harmonic functions to higher dimensions.

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11

Mateljevic, Miodrag, and Marek Svetlik. "Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings." Applicable Analysis and Discrete Mathematics 14, no. 1 (2020): 150–68. http://dx.doi.org/10.2298/aadm200104001m.

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We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasiregular, shortly HQR) mappings with the strip codomain. Along the way, we get a simple proof of a new version of the Schwarz lemma for real valued harmonic functions (without the assumption that 0 is mapped to 0 by the corresponding map). Using the Schwarz-Pick lemma related to distortion for harmonic functions and the elementary properties of the hyperbolic geometry of the strip we get optimal estimates for modulus of HQR mappings.

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12

Pal, Sourav, and Samriddho Roy. "A generalized Schwarz lemma for two domains related to μ-synthesis." Complex Manifolds 5, no. 1 (February 2, 2018): 1–8. http://dx.doi.org/10.1515/coma-2018-0001.

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AbstractWe present a set of necessary and sufficient conditions that provides a Schwarz lemma for the tetrablock E. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc G2. In either case, our results generalize all previous results in this direction for E and G2.

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13

Hamada, Hidetaka. "A Schwarz lemma on complex ellipsoids." Annales Polonici Mathematici 67, no. 3 (1997): 269–75. http://dx.doi.org/10.4064/ap-67-3-269-275.

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14

Krantz, Steven G. "The Schwarz lemma at the boundary." Complex Variables and Elliptic Equations 56, no. 5 (May 2011): 455–68. http://dx.doi.org/10.1080/17476931003728438.

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15

Örnek, Nafi, and Burcu Gök. "Boundary Schwarz lemma for holomorphic functions." Filomat 31, no. 18 (2017): 5553–65. http://dx.doi.org/10.2298/fil1718553o.

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In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function f (z) holomorphic in the unit disc and f (0) = 0 such that ?Rf? < 1 for ?z? < 1, we estimate a modulus of angular derivative of f (z) function at the boundary point b with f (b) = 1, by taking into account their first nonzero two Maclaurin coefficients. Also, we shall give an estimate below ?f'(b)? according to the first nonzero Taylor coefficient of about two zeros, namely z=0 and z0 ? 0. Moreover, two examples for our results are considered.

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16

Knese, Greg. "A Schwarz lemma on the polydisk." Proceedings of the American Mathematical Society 135, no. 09 (March 30, 2007): 2759–69. http://dx.doi.org/10.1090/s0002-9939-07-08766-7.

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17

Klimek, M. "Infinitesimal pseudometrics and the Schwarz lemma." Proceedings of the American Mathematical Society 105, no. 1 (January 1, 1989): 134. http://dx.doi.org/10.1090/s0002-9939-1989-0930248-4.

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18

Mackey, M., and P. Mellon. "A Schwarz Lemma and Composition Operators." Integral Equations and Operator Theory 48, no. 4 (April 1, 2004): 511–24. http://dx.doi.org/10.1007/s00020-003-1240-1.

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19

Dineen, Seán, and Richard M. Timoney. "Extremal mappings for the Schwarz lemma." Arkiv för Matematik 30, no. 1-2 (December 1992): 61–81. http://dx.doi.org/10.1007/bf02384862.

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20

Beardon, A. F., and D. Minda. "A multi-point Schwarz-Pick Lemma." Journal d'Analyse Mathématique 92, no. 1 (December 2004): 81–104. http://dx.doi.org/10.1007/bf02787757.

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21

Zhang, Zhongxiang. "The Schwarz lemma in Clifford analysis." Proceedings of the American Mathematical Society 142, no. 4 (January 6, 2014): 1237–48. http://dx.doi.org/10.1090/s0002-9939-2014-11854-5.

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22

Beardon, A. F. "The Schwarz-Pick Lemma for derivatives." Proceedings of the American Mathematical Society 125, no. 11 (1997): 3255–56. http://dx.doi.org/10.1090/s0002-9939-97-03906-3.

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23

Ito, Manabu. "Schwarz Lemma in infinite-dimensional spaces." Monatshefte für Mathematik 191, no. 4 (January 29, 2020): 735–48. http://dx.doi.org/10.1007/s00605-020-01375-x.

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24

Liu, Bingyuan. "Two applications of the Schwarz lemma." Pacific Journal of Mathematics 296, no. 1 (May 1, 2018): 141–53. http://dx.doi.org/10.2140/pjm.2018.296.141.

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25

Mercer, Peter R. "Sharpened Versions of the Schwarz Lemma." Journal of Mathematical Analysis and Applications 205, no. 2 (January 1997): 508–11. http://dx.doi.org/10.1006/jmaa.1997.5217.

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26

Janušauskas, A. "Generalization of Holmgren's lemma." Lithuanian Mathematical Journal 31, no. 4 (October 1991): 501–3. http://dx.doi.org/10.1007/bf00970800.

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27

KALAJ, DAVID. "SCHWARZ LEMMA FOR HOLOMORPHIC MAPPINGS IN THE UNIT BALL." Glasgow Mathematical Journal 60, no. 1 (September 4, 2017): 219–24. http://dx.doi.org/10.1017/s0017089517000052.

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AbstractIn this note, we establish a Schwarz–Pick type inequality for holomorphic mappings between unit balls Bn and Bm in corresponding complex spaces. We also prove a Schwarz-Pick type inequality for pluri-harmonic functions.

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28

Klimek, M. "Infinitesimal Pseudo-Metrics and the Schwarz Lemma." Proceedings of the American Mathematical Society 105, no. 1 (January 1989): 134. http://dx.doi.org/10.2307/2046747.

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29

Mercer, Peter R. "Boundary Schwarz inequalities arising from Rogosinski's lemma." Journal of Classical Analysis, no. 2 (2018): 93–97. http://dx.doi.org/10.7153/jca-2018-12-08.

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30

Jeong, Moon-Ja. "THE SCHWARZ LEMMA AND BOUNDARY FIXED POINTS." Pure and Applied Mathematics 18, no. 3 (August 31, 2011): 275–84. http://dx.doi.org/10.7468/jksmeb.2011.18.3.275.

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31

AKYEL, TUGBA, and NAFI ORNEK. "A SHARP SCHWARZ LEMMA AT THE BOUNDARY." Pure and Applied Mathematics 22, no. 3 (August 31, 2015): 263–73. http://dx.doi.org/10.7468/jksmeb.2015.22.3.263.

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32

Verma, K. "A Schwarz lemma for correspondences and applications." Publicacions Matemàtiques 47 (July 1, 2003): 373–87. http://dx.doi.org/10.5565/publmat_47203_04.

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33

Kalaj, David, and Matti Vuorinen. "On harmonic functions and the Schwarz lemma." Proceedings of the American Mathematical Society 140, no. 1 (May 2, 2011): 161–65. http://dx.doi.org/10.1090/s0002-9939-2011-10914-6.

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34

Agler, J., and N. J. Young. "A Schwarz Lemma for the Symmetrized Bidisc." Bulletin of the London Mathematical Society 33, no. 2 (March 2001): 175–86. http://dx.doi.org/10.1112/blms/33.2.175.

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35

Chelst, Dov. "A generalized Schwarz lemma at the boundary." Proceedings of the American Mathematical Society 129, no. 11 (June 6, 2001): 3275–78. http://dx.doi.org/10.1090/s0002-9939-01-06144-5.

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36

Cheung, Leung-Fu, and Pui-Fai Leung. "A Schwarz lemma for complete Riemannian manifolds." Bulletin of the Australian Mathematical Society 55, no. 3 (June 1997): 513–15. http://dx.doi.org/10.1017/s000497270003416x.

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We prove a Schwarz Lemma for conformal mappings between two complete Riemannian manifolds when the domain manifold has Ricci curvature bounded below in terms of its distance function. This gives a partial result to a conjecture of Chua.

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37

Cho, Kyung Hyun, Seong-A. Kim, and Toshiyuki Sugawa. "On a Multi-Point Schwarz-Pick Lemma." Computational Methods and Function Theory 12, no. 2 (August 21, 2012): 483–99. http://dx.doi.org/10.1007/bf03321839.

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38

Beardon, Alan F., and Kenneth Stephenson. "The Schwarz-Pick Lemma for circle packings." Illinois Journal of Mathematics 35, no. 4 (December 1991): 577–606. http://dx.doi.org/10.1215/ijm/1255987673.

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39

Mishra, Akshaya Kumar. "Some applications of Schwarz Lemma for operators." International Journal of Mathematics and Mathematical Sciences 12, no. 2 (1989): 349–53. http://dx.doi.org/10.1155/s0161171289000402.

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40

Mercer, Peter R. "An improved Schwarz Lemma at the boundary." Open Mathematics 16, no. 1 (October 19, 2018): 1140–44. http://dx.doi.org/10.1515/math-2018-0096.

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41

Savas-Halilaj, Andreas. "A Schwarz–Pick lemma for minimal maps." Annals of Global Analysis and Geometry 56, no. 2 (May 16, 2019): 193–201. http://dx.doi.org/10.1007/s10455-019-09663-y.

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42

Bernal-González, L., and M. C. Calderón-Moreno. "Two hyperbolic Schwarz lemmas." Bulletin of the Australian Mathematical Society 66, no. 1 (August 2002): 17–24. http://dx.doi.org/10.1017/s0004972700020633.

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In this paper, a sharp version of the Schwarz–Pick Lemma for hyperbolic derivatives is provided for holomorphic selfmappings on the unit disk with fixed multiplicity for the zero at the origin. This extends a recent result due to Beardon. A property of preserving hyperbolic distances also studied by Beardon is here completely characterised.

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43

Chen, HuaiHui. "The Schwarz-Pick lemma and Julia lemma for real planar harmonic mappings." Science China Mathematics 56, no. 11 (August 19, 2013): 2327–34. http://dx.doi.org/10.1007/s11425-013-4691-0.

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44

MOHAPATRA, MANAS RANJAN, XIANTAO WANG, and JIAN-FENG ZHU. "BOUNDARY SCHWARZ LEMMA FOR SOLUTIONS TO NONHOMOGENEOUS BIHARMONIC EQUATIONS." Bulletin of the Australian Mathematical Society 100, no. 3 (September 9, 2019): 470–78. http://dx.doi.org/10.1017/s0004972719000947.

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45

Kwon, Ern, Jinkee Lee, Gun Kwon, and Mi Kim. "A Refinement of Schwarz–Pick Lemma for Higher Derivatives." Mathematics 7, no. 1 (January 13, 2019): 77. http://dx.doi.org/10.3390/math7010077.

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In this paper, a Schwarz–Pick estimate of a holomorphic self map f of the unit disc D having the expansion f ( w ) = c 0 + c n ( w − z ) n + … in a neighborhood of some z in D is given. This result is a refinement of the Schwarz–Pick lemma, which improves a previous result of Shinji Yamashita.

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46

BERINDE, VASILE. "A generalization of Mortici lemma." Creative Mathematics and Informatics 21, no. 2 (2012): 129–34. http://dx.doi.org/10.37193/cmi.2012.02.02.

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The aim of this note is to obtain a generalization of a very simple, elegant but powerful convergence lemma introduced by Mortici [Mortici, C., Best estimates of the generalized Stirling formula, Appl. Math. Comp., 215 (2010), No. 11, 4044–4048; Mortici, C., Product approximations via asymptotic integration, Amer. Math. Monthly, 117 (2010), No. 5, 434–441; Mortici, C., An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel), 93 (2009), No. 1, 37–45; Mortici, C., Complete monotonic functions associated with gamma function and applications, Carpathian J. Math., 25 (2009), No. 2, 186–191] and exploited by him and other authors in an impressive number of recent and very recent papers devoted to constructing asymptotic expansions, accelerating famous sequences in mathematics, developing approximation formulas for factorials that improve various classical results etc. We illustrate the new result by some important particular cases and also indicate a way for using it in similar contexts.

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47

Asch, J., and J. Potthoff. "A generalization of Itô's lemma." Proceedings of the Japan Academy, Series A, Mathematical Sciences 63, no. 8 (1987): 289–91. http://dx.doi.org/10.3792/pjaa.63.289.

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48

Ait Mansour, M., M. A. Bahraoui, and A. El Bekkali. "A generalization of Lim's lemma." Journal of Nonlinear Sciences and Applications 14, no. 01 (June 13, 2020): 48–53. http://dx.doi.org/10.22436/jnsa.014.01.06.

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49

Panyanak, B., and A. Cuntavepanit. "A Generalization of Suzuki's Lemma." Abstract and Applied Analysis 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/824718.

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Let{zn},{wn}, and{vn}be bounded sequences in a metric space of hyperbolic type(X,d), and let{αn}be a sequence in[0,1]with0<lim⁡⁡inf⁡n⁡αn≤lim⁡⁡sup⁡n⁡αn<1. Ifzn+1=αnwn⊕(1−αn)vnfor alln∈ℕ,lim⁡n⁡d(zn,vn)=0, andlim⁡⁡sup⁡n⁡(d(wn+1,wn)-d(zn+1,zn))≤0, thenlim⁡n⁡d(wn,zn)=0. This is a generalization of Lemma 2.2 in (T. Suzuki, 2005). As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces.

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50

Haussler, David, and Philip M. Long. "A generalization of Sauer's lemma." Journal of Combinatorial Theory, Series A 71, no. 2 (August 1995): 219–40. http://dx.doi.org/10.1016/0097-3165(95)90001-2.

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