Shimomura, Shun Growth of modified Painlevé transcendents of the fifth and the third kind. (English) Zbl 1058.34127 Forum Math. 16, No. 2, 231-247 (2004). The author considers alternative versions of the third and fifth Painlevé equations. His aim is to estimate the magnitude of the characteristic function \(T(r,f)\) of the Nevanlinna theory for solutions of these two equations. He also comments on the classical work of P. Boutroux [Acta Math. 28, 97–224 (1904; JFM 35.0411.02)] in the case of the alternative version of the third Painlevé equation. Reviewer: Andrew Pickering (Madrid) Cited in 6 Documents MSC: 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 33E17 Painlevé-type functions Keywords:third and fifth Painlevé transcendents; Nevanlinna theory Citations:JFM 35.0411.02 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Boutroux P., Acta Math. 28 pp 97– (1904) [2] Garnier R., J. Math. Pures Appl. 46 pp 353– (1967) [3] Gromak V. I.: Theory of Painleve’s equation. Di erentsial’nye Uravneniya 11 (1975), 373-376 [Russian], Engl. Transl.: Di erential Equations 11 (1975), 285-287 [4] Gromak V. I.: Solutions of Painleve’s fifth equation. Di erentsial’nye Uravneniya 12 (1976), 740-742 [Russian], Engl. Transl.: Di erential Equations 12 (1976), 519-521 [5] Gromak V. I.: Reducibility of the Painleve equations. Di erentsial’nye Uravneniya 20 (1984), 1674-1683 [Russian], Engl. Transl.: Di erential Equations 20 (1984), 1191-1198 [6] Hayman W. K.: Meromorphic Functions. Clarendon Press, Oxford 1964 · Zbl 0115.06203 [7] Hille E.: Ordinary Di erential Equations in the Complex Domain. Wiley, New York 1976 · Zbl 0343.34007 [8] Hinkkanen A., J. Anal. Math. 85 pp 323– (2001) [9] Hinkkanen A., Report. Univ. Jyvskyl 83 pp 133– (2001) [10] Jank G. and Volkmann L.: Einf hrung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Di erentialgleichungen. Birkh user, Basel 1985 · Zbl 0682.30001 [11] Kitaev A. V., Law C. K. and McLeod J. B.: Rational solutions of the fifth Painleve equation. Di erential Integral Equations 7 (1994), 967-1000 · Zbl 0804.34006 [12] Laine I.: Nevanlinna Theory and Complex Di erential Equations. de Gruyter, Berlin New York 1993 [13] Lukashevich N. A.: Solutions of the fifth equation of Painleve. Di erentsial’nye Uravneniya 4 (1968), 1413-1420 [Russian], Engl. Transl.: Di erential Equations 4 (1968), 732-735 [14] Murata Y., Nagoya Math. J. 139 pp 37– (1995) [15] Okamoto K., Funkcial. Ekvac. 44 pp 201– (2001) [16] Painleve P. uvres de Paul Painleve, tome III. Centre National de la Recherche Scientifiques, Paris 1975 [17] Shimomura S., Complex Variables 40 pp 51– (1999) [18] Shimomura S., Result. Math. 38 pp 348– (2000) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.