Value distribution of a Wronskian (Q1888641)

This is an interesting paper in value distribution theory, making use of the integrated version, introduced by\textit{N. Toda}, of the basic Nevanlinna theory functions, see [Tohoku Math. J., II. Ser. 22, 635--658 (1970; Zbl 0213.09202)]. The main results in this paper are as follows: (1) Given a transcendental meromorphic function \(f\) with the maximal deficiency sum, let \(a_1\equiv1\), \(a_2,\dots,a_k\) be linearly independent meromorphic functions which are small in the sense that \(T(cr,a_j)=o(T(r,f))\) as \(r\to\infty\) for some \(c=c_j>1\). Denoting by \(L(f)\) the Wronskian determinant \(W(a_1,a_2,\dots,a_k,f)\), then \[ \lim_{r\to\infty}\frac{N(r,L(f))+N(r,1/L(f))}{T(r,L(f))} =\frac{2k(1-\delta(\infty,f))}{1+k-k\delta(\infty,f)}. \] This is a generalization of previous results due to \textit{S. K. Singh} and \textit{V. V. Kulkarni} [Ann. Pol. Math. 28, 123--133 (1973; Zbl 0273.30025)] and to \textit{M. Fang} [Int. J. Math. Math. Sci. 23, No. 4, 285--288 (2000; Zbl 0954.30018)]. (2) Given a transcendental meromorphic function \(f\) with the maximal deficiency sum such that \(\delta(\infty,f)=0\), then for any natural number \(k\), \[ \sum_{b\neq\infty}\delta(b,f^{(k)})=\delta(0,f^{(k)})=\frac2{k+1}. \] This result extends a previous result due to \textit{L. Yang} and \textit{Y. Wang} [Sci. China, Ser. A 35, No. 10, 1180--1190 (1992; Zbl 0760.30011)] to the functions \(f\) of infinite order, see also the original problem posed by \textit{D. Drasin} in [Complex Anal. Appl., int. Summer Course Trieste 1975, Vol. I, 1--93 (1976; Zbl 0345.31002)].