A simpler proof of a theorem of Steinmetz (Q1824724)

Let \(f_ j,h_ j\not\equiv 0\) be meromorphic functions and let g be a nonconstant entire function such that \(\sum^{n}_{j=1}f_ j(g)h_ j=0\) and \(\sum^{n}_{j=1}T(r,h_ j)=O(T(r,g))\) for arbitrarily large r. Then there exist nontrivial polynomials \(p_ j\) such that \(\sum^{n}_{j=1}p_ j(g)h_ j=0\). The authors of the present paper give a new proof of this theorem, which makes use of Nevanlinna's First Main Theorem, but avoids his Second Main Theorem. Some recent applications to factorization theory and fixed points of composite functions are pointed out.