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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.

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D45 Normal functions of one complex variable, normal families
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