Yang, Chung-Chun; Ye, Zhuan Estimates of the proximate function of differential polynomials. (English) Zbl 1122.30022 Proc. Japan Acad., Ser. A 83, No. 4, 50-55 (2007). Summary: We obtain a Clunie type theorem for a rather general form of functional equations involving differential polynomials. Our theorems can give a much sharper estimate on the error term of the proximity function of solutions of differential equations and functional equations than the upper bound obtained by Clunie, Doeringer, He-Xiao, Korhonen and etc. In particular, our theorem can also be applied to study various types of Painlevé differential equations. Cited in 18 Documents MSC: 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies Keywords:Nevanlinna theory; differential polynomial; Painlevé equations × Cite Format Result Cite Review PDF Full Text: DOI Euclid References: [1] W. Cherry and Z. Ye, Nevanlinna’s theory of value distribution , Springer, Berlin, 2001. · Zbl 0981.30001 [2] J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17-27. · Zbl 0104.29504 · doi:10.1112/jlms/s1-37.1.17 [3] W. Doeringer, Exceptional values of differential polynomials, Pacific J. Math. 98 (1982), no. 1, 55-62. · Zbl 0445.30025 · doi:10.2140/pjm.1982.98.55 [4] A. A. Gol’dberg and V. A. Grinšteĭ n, The logarithmic derivative of a meromorphic function, Mat. Zametki 19 (1976), no. 4, 525-530. [5] Y. Z. He and X. Z. Xiao, Algeborid functions and ordinary differential equations , Science Press, Beijing, 1988. [6] R. Korhonen, Sharp forms of Nevanlinna error terms in differential equations, in Symposium on Complex Differential and Functional Equations , Univ. Joensuu, Joensuu, 2004, pp. 117-133, Erratum: http://www.arxiv.org/abs/ math.CV/0608514, 2006, pp. 15-16. · Zbl 1154.30305 [7] I. Laine, Nevanlinna theory and complex differential equations , de Gruyter, Berlin, 1993. [8] A. A. Mohon’ko and V. D. Mohon’ko, Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions, and their applications to differential equations, Sibirsk. Mat. Ž. 15 (1974), 1305-1322, 1431. · Zbl 0305.30029 · doi:10.1007/BF00966560 [9] S. Shimomura, The first, the second and the fourth Painlevé transcendents are of finite order, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 3, 42-45. · Zbl 1097.34065 · doi:10.3792/pjaa.77.42 [10] S. Shimomura, Lower estimates for the growth of Painlevé transcendents, Funkcial. Ekvac. 46 (2003), no. 2, 287-295. · Zbl 1162.34369 · doi:10.1619/fesi.46.287 [11] S. Shimomura, Growth of the first, the second and the fourth Painlevé transcendents, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 2, 259-269. · Zbl 1030.34086 · doi:10.1017/S0305004102006400 [12] N. Steinmetz, Global properties of the Painlevé transcendents: new results and open questions, Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 1, 71-98. · Zbl 1081.34088 [13] N. Steinmetz, Value distribution of the Painlevé transcendents, Israel J. Math. 128 (2002), 29-52. · Zbl 1016.34091 · doi:10.1007/BF02785417 [14] Z. Ye, On Nevanlinna’s second main theorem in projective space, Invent. Math. 122 (1995), no. 3, 475-507. · Zbl 0855.32002 · doi:10.1007/BF01231453 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.