Growth and value distribution of differential monomials (Q5960346)

Let \(f(z)\) denote a transcendental meromorphic function in \(\mathbb{C}\). The authors study the growth and the value distribution of differential monomials for simple zeros. For \(a\in\overline\mathbb{C}\) they denote by \(n(r,a; f|=1)\) the number of simple zeros of \(f(z)-a\) in \(|z|\leq r\), and define \(N(r,a;f |=1)\) in the usual way. They set \(\delta_1(a;f)=1-\lim\sup_{r\to \infty}N (r,a;f|= 1)/T(r,f)\), where \(T(r,f)\) is the characteristic function of \(f(z)\). \textit{H. X. Yi} [Bull. Austral. Math. Soc. 41, No. 3, 417-420 (1990; Zbl 0693.30025)] proved that if \(\sum_{\alpha\in \overline\mathbb{C}} \delta_1(a;f)=4\) then \(\lim_{r\to\infty} T(r,f^{(k)})/T(r,f)= k+1-(k/2) \delta_1(\infty,f)\) under the condition \(f(z)\) is of finite order. The authors investigate this problem replacing the \(f^{(k)}\) by a differential monomial \(P[f]\) and relaxing the order restriction in the result above. They proved that if \(\sum_{a\in \overline\mathbb{C}}\delta_1 (a;f)=4\), then \(\delta (\infty, P[f])=0\), and for any \(b\neq\infty\), \(\delta(0,f)\leq 1-\Theta(b;f)/(2-\Theta(\infty;f))\).