Zero distribution of some shift polynomials
Let \(f\) be a meromorphic function of finite order \(\rho\). A meromorphic function \(\alpha\) is defined as small relative to \(f\) if for any \(\varepsilon > 0\) and for some \(\lambda < \rho\), \(T(r, \alpha) = O(r^{\lambda + \varepsilon}) + S(r, f)\), possibly outside a set of finite logarithmic measure. Let \(c_{1}, c_{2}, \ldots , c_{k}\) be nonzero distinct complex constants and \(b_{0}(z), b_{1}(z), \ldots , b_{k}(z)\) be small meromorphic functions relative to \(f\). The author defines a shift polynomial as \[ g(f) = \sum_{j = 1}^{k} b_{j}(z) f(z + c_{j}). \] He proves three results on the value distribution of the functions of the form \(F = f^{n}(g(f))^{s} - b_{0}\), for different values of \(n\) and \(s\). He establishes some relations between the exponent of convergence of the zeros of the shift polynomial and the order of \(f\).
scientific article; zbMATH DE number 6958148
scientific article