The value distribution of entire functions of finite order
The author proves the following theorem: Let \(f(z)\) be an entire function and let \(\{w_ n\}\) be an unbounded sequence. Suppose that for some positive integer \(m\) \[ \underline{\lim}_{r\to \infty}\frac{T(r,f)}{r^ m}=0. \] Assume that there exists some \(\varepsilon >0\) such that all the roots of \( f(z)=w_ n\) \((n=1,2,...) \) belong to the following set \[ \cup^{m-1}_{k=0}\{z;\frac{2k}{m}\pi +\varepsilon <\arg z<\frac{2k+1}{m}\pi -\varepsilon \}. \] Then \(f(z)\) is a polynomial.
scientific article; zbMATH DE number 4130730