Valiron deficient functions of meromorphic functions: An extension (Q1342492)

A meromorphic function \(\varphi\) is a Valiron deficient function of a meromorphic function \(f\) if \(T(r, \varphi)= o(T (r,f))\) are the deficiency \[ \Delta (\varphi)= \varlimsup_{r\to \infty} {{m \left( r, {1\over {f- \varphi}} \right)} \over {T(r,f)}} >0. \] Let \(c(f)\) be the first non-zero coefficient of the Laurent development centered at zero of a meromoprhic function \(f\) and let \[ d(f, g)= \begin{cases} e^{c(f -g)}, &g\not\equiv g\\ 0, &g\equiv g \end{cases} . \] A set \({\mathcal F}\) of meromorphic functions is called a \(\mu\)-set if there exist a number \(\sigma>0\) and a sequence \(\varphi_n\) of meromorphic functions such that \[ {\mathcal F} \subset \bigcap_{N=1}^\infty\;\bigcup_{n=N}^\infty \bigl\{ \varphi:\;d(\varphi, \varphi_n)< \exp (-\sigma n) \bigr\}. \] Let \(\Phi= \{\Phi_1, \Phi_2, \dots\}\) be a sequence of meromorphic functions such that the system \(\Phi_1, \dots, \Phi_n\) is linearly independent for every \(n\), \(M(\Phi)\) be a linear span of \(\Phi\). The main result of the article is the following: Let \(f(z)\) be a transcendental meromorphic function of finite order, \(T(r, \Phi_j)= o(T(r, f))\), \(j=1, 2,\dots\), and let \({\mathcal F}_\delta= \{\varphi\in M(\Phi)\): \(\Delta (\varphi)> \delta\}\), \(0< \delta<1\); then \({\mathcal F}_{\delta}\) is a \(\mu\)-set. The case \(\Phi_j\) being entire functions was studied by Chen (1983).