On the growth of meromorphic functions (Q5896545)

Let f be a transcendental meromorphic function in the plane. We denote by Q[f]\(\not\equiv 0\) a differential polynomial of f with small coefficients and by \(\Gamma\) the weight of Q[f]. Furthermore we define by \(P(f)=(f^ n+a_{n-1}f^{n-1}+...+a_ 0)^ k\) a polynomial in f with small coefficients \(a_ j\) and k,n\(\in {\mathbb{N}}\) with \(k\geq 3\). Finally we define by \(F=P(f)\cdot Q[f]\) and assume that \[ K:=(k-2)n-(2- \frac{3}{n})\Gamma \] is positive. Then the authors prove that \[ T(r,f)<\frac{1}{K}\bar N(r,\frac{1}{F-\phi})+S(r,f), \] where \(\phi\not\equiv 0\) is an arbitrary small meromorphic function with respect to f.