Defective functions of meromorphic functions in the unit disc (Q289623)

The second fundamental theorem due to Nevanlinna for meromorphic functions in the complex plane has been generalized to moving targets. Concerning the defect relation, \textit{G. Frank} and \textit{G. Weissenborn} [Bull. Lond. Math. Soc. 18, 29--33 (1986; Zbl 0586.30025)] obtained an analogue for rational functions as targets. For the case of general small functions as targets, \textit{N. Steinmetz} solved this problem in [J. Reine Angew. Math. 368, 134--141 (1986; Zbl 0598.30045)]. The tools are auxiliary functions defined by Wronski determinants. NEWLINENEWLINEThe author considers the unit disc counterpart of the defect relations. The following is shown: Let \(f\) be a meromorphic function in the unit disc and let \(a_1,\dots,a_k\) be distinct small meromorphic functions with respect to \(f\). Then, for any \(\varepsilon>0\) the following inequality holds NEWLINE\[NEWLINEm(r,f)+\sum\limits_{\nu=1}^k m(r,a_\nu,f)\leq (2+\varepsilon)T(r,f)+O\left(\log\frac1{1-r}\right)+S(r,f).NEWLINE\]NEWLINE In the proof, some types of Wronski determinants play an important role.NEWLINENEWLINEAs the author mentions, for meromorphic functions in the complex plane, \textit{K. Yamanoi} obtained the exact analogue of the second main theorem including the ramification relation [Acta Math. 192, No. 2, 225--294 (2004; Zbl 1203.30035)].NEWLINENEWLINEThe author also considers analogues of Ullrich's theorem for small functions in the unit disc.