Relative defects for homogeneous differential polynomials (Q2716772)

This paper is devoted to considering several types of defects for homogeneous differential polynomials. In addition to the usual defect notion \(\delta(\alpha, f)\), relative defects such as NEWLINE\[NEWLINE \delta_{r}^{(k)}(\alpha, f):=1-\limsup_{t\to\infty} \Big(N\big(t,1/(f^{(k)}-\alpha)\big)/T(t,f)\Big), NEWLINE\]NEWLINE NEWLINE\[NEWLINE \delta_{r}^{(k)}(\alpha, P_{n}(f)):=1-\limsup_{t\to\infty} \Big(N\big(t,1/(P_{n}(f)^{(k)}-\alpha)\big)/T(t,f)\Big), NEWLINE\]NEWLINE and corresponding notions for distinct zeros will be considered; here \(P_{n}(t)\) stands for a differential polynomial of total degree \(n\) with meromorphic coefficients \(a(z)\) of type \(T(t,a(z))=S(t,f)\). Two examples of typical results read as follows: NEWLINENEWLINENEWLINE(1) For a homogeneous differential polynomial \(D_{n}(f)\) of a meromorphic function \(f\) of total degree \(n\) and for \(\alpha\in{\mathbf C}\), \(\Theta_{r}^{(k)}(\alpha, D_{n})\leqq 2-\Theta(\infty,f)-n\delta(0,f)\). NEWLINENEWLINENEWLINE(2) For a differential polynomial as in (1), but not containing \(f\) explicitly, for distinct, non-zero complex numbers \(a_{1},\ldots,a_{p}\), and for non-zero complex number \(b_{1},\ldots,b_{p}\), NEWLINE\[NEWLINE\begin{multlined}\sum_{i=1}^{p}\Theta(a_{i},f)+\Theta(0,f)+2\Theta(\infty,f) +\sum_{j=1}^{q}\Theta_{r}^{(k)}(b_{j},D_{n})+\Theta_{r}^{(k)}(0,D_{n}) \\ \leqq 4+q\big(\delta_{r}^{(k)})(0,D_{n})-n\delta(0,f)\big). \end{multlined}NEWLINE\]NEWLINE Several misprints disturb reading the paper.