A modification of the Nevanlinna theory
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 [\textit{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 \textit{H. Cartan} [Annales Ecole norm. (3) 45, 255--346 (1928; JFM 54.0357.06)] and \textit{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 \textit{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.
scientific article; zbMATH DE number 5723302
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