Value distribution of certain differential polynomials of algebroid functions (Q2365011)

We obtain the following results: (i) Let \(w(z)\) be a \(\nu\)-valued transcendental algebroid function and set \(\varphi(z): =w'(z) - aw(z)^n\), where \(n\in \mathbb{N}\) nad \(a\in \mathbb{C} \backslash \{0\}\). Then if \(n\geq 4\nu + 1\), we have \[ \overline N \left(r,{1 \over \varphi-b} \right) \neq S(r,w) \] for each \(b\in \mathbb{C}\). (ii) Let \(w(z)\) be a \(\nu\)-valued transcendental algebroid function and set \(\varphi(z): =w'(z) w(z)^n\), where \(n\in \mathbb{N}\). Then if \(n\geq 4\nu-1\), we have \[ \overline N \left(r,{1 \over \varphi-b} \right) \neq S(r,w) \] for each \(b\in \mathbb{C} \backslash \{0\}\).