Proximate deficiency of homogeneous differential polynomials (Q1917743)

Let \(f\) be a transcendental meromorphic function in the complex plane. For \(j\geq 1\), let \(n_{0j},n_{1j},\dots,n_{kj}\) \((k\geq1)\) be nonnegative integers with \(\sum^k_{i=0} n_{ij}\geq1\), and let \[ M_j=A_j(f)^{n_{0j}}(f')^{n_{1j}}\cdots(f^{(k)})^{n_{kj}} \] with \(A_j\) a meromorphic function in the plane for which \(T(r,A_j)=S(r,f)\) in the usual notation of Nevanlinna. If \(f\) has finite order \(\rho\), a proximate order \(\rho_f(r)\) exists for \(f\) and following Valiron the quantity \[ \delta_{\rho_f}(a,f)=1-\limsup_{r\to\infty} \{N(r,a)/r^\rho f^{(r)}\} \] is called the proximate deficiency of the complex value \(a\). The authors prove a number of results concerning proximate deficiencies for differential polynomials \(P[f]=\sum^l_{j=1} M_j\) generated by \(f\).