×

zbMATH — the first resource for mathematics

A modification of the Nevanlinna theory. (English) Zbl 1207.30047
This paper presents a modification of the Nevanlinna theory by making use of the full generality of the Poisson–Jensen formula instead of using a special case of the formula, the Jensen formula, that had been used to deduce the “classical” Nevanlina theory. More precisely, let \(\alpha\) be a point in the disk \(|z|<R_{0}\leq\infty\) and suppose that \(f(\alpha )\neq\infty\) for a given meromorphic function \(f\). Then for \(\alpha <r<R_{0}\), define the modified Nevanlinna functions as \[ m_{\alpha}(r,f)=\frac{1}{2\pi}\int_{0}^{2\pi}\log^{+}|f(re^{it})| Re \frac{re^{it}+\alpha}{re^{it}-\alpha}dt, \]
\[ N_{\alpha}(r,f)=\sum_{|b_{k}|<r}\log \left| \frac{r^{2}-\overline{b_{k}}\alpha}{r(\alpha -b_{k})}\right|, \]
\[ T_{\alpha}(r,f)=m_{\alpha}(r,f)+N_{\alpha}(r,f), \] where the \(b_{k}\) stand for the poles of \(f\). Then, several standard Nevanlinna theory results follow in terms of these modified functions, including the first and second main theorems, the Cartan formula, and a variant of the logarithmic derivative lemma. The proofs are, in fact, natural modifications of the corresponding “classical” results. As pointed out by the author, these modified functions have been applied previously by Nevanlinna himself, see [R. Nevanlinna, Analytic functions. Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften. 162. Berlin-Heidelberg-New York: Springer-Verlag. VIII, 373 p. (1970; Zbl 0199.12501)], who used them to study meromorphic functions of bounded type. Somewhat surprisingly, the present modified theory has not been completely appeared before, although similar developments are due to H. Cartan [Annales Ecole norm. (3) 45, 255–346 (1928; JFM 54.0357.06)] and D. Drasin [Acta Math., 122, 231–263 (1969; Zbl 0176.02802)]. As an example of possible applications, the author proves an estimate for the modified proximity function of certain differential polynomials, see W. Doeringer [Pac. J. Math., 98, 55–62 (1982; Zbl 0445.30025)] for a corresponding “classical” result. The author also proposes to treat differential-difference polynomials in terms of the modified theory, to appear in a forth-coming paper. The reviewer is confident of the importance of the present paper, to be applied for a number of applications in the field of value distribution theory.

MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D45 Normal functions of one complex variable, normal families
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] W. Bergweiler, Bloch’s principle, Comput. Methods Funct. Theory 6 (2006), 77–108. · Zbl 1101.30034
[2] H. Cartan, Sur les systèmes de fonctions holomorphes a variétés linéaires lacunaires et leurs applications, Ann. École Norm. Sup. (3) 45 (1928), 255–346. · JFM 54.0357.06
[3] W. Döringer, Exceptional values of differential polynomials, Pacific J. Math. 98 (1982), 55–62 · Zbl 0469.30023
[4] D. Drasin, Normal families and the Nevanlinna theory, Acta Math. 122 (1969), 231–263. · Zbl 0176.02802
[5] J. Grahl, Some applications of Cartan’s theorem to normality and semiduality of gap power series, J. Anal. Math. 82 (2000), 207–220. · Zbl 0970.30003
[6] J. Grahl, Hayman’s alternative and normal families of nonvanishing meromorphic functions, Comput. Methods Funct. Theory 2 (2002), 481–508. · Zbl 1062.30033
[7] –, Differential polynomials with dilations in the argument and normal families, preprint. · Zbl 1221.30077
[8] W. K. Hayman, Meromorphic Functions, Oxford University Press, London, 1964.
[9] G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser, Basel Boston Stuttgart, 1985. · Zbl 0682.30001
[10] R. Nevanlinna, Analytic Functions, Springer, Berlin, 1970. · Zbl 0199.12501
[11] X. Pang, Bloch’s principle and normal criterion, Sci. Sinica 32 no.7 (1989), 782–791. · Zbl 0687.30023
[12] X. Pang, On normal criterion of meromorphic functions, Sci. Sinica 33 no.5 (1990), 521–527 · Zbl 0706.30024
[13] St. Ruscheweyh and L. Salinas, On some cases of Bloch’s principle, Scientia Ser. A, Math. Sciences 1 (1988), 97–100. · Zbl 0685.30026
[14] St. Ruscheweyh and K.-J. Wirths, Normal families of gap power series, Results in Math. 10 (1986), 147–151. · Zbl 0623.30044
[15] J. Schiff, Normal Families, Springer, New York, 1993.
[16] W. Schwick, Normality criteria for families of meromorphic functions, J. Anal. Math. 52 (1989), 241–289. · Zbl 0667.30028
[17] W. Schwick, An estimation of the proximity function of the logarithmic derivative for families of meromorphic functions, Compl. Var. 15 (1990), 149–154. · Zbl 0728.30024
[18] L. Yang, Value Distribution Theory, Springer, Berlin Heidelberg, 1993. · Zbl 0790.30018
[19] L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813–817. · Zbl 0315.30036
[20] L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. 35 (1998), 215–230. Jürgen Grahl · Zbl 1037.30021
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.