An algebroid function and its derivative
Let some of the functions \[ A_v(z),\;A_{v-1}(z),\dots,A_0(z) \] be transcendental entire functions. Then the relation \[ A_v(z)W^v+A_{v-1} (z)W^{v-1}+\cdots+A)1(z) W+A_0(z)=0 \] defines a \(v\)-valued transcendental algebroid function \(W\) on the complex plane. The author takes for granted that the reader of his paper is familiar with Nevanlinna's value distribution theory. Using these results, the author proves, for example, the following relation for the characteristic functions \(T(r,W)\) and \(T(r,W')\) of \(W\) and its derivative \(W'\): Theorem 3.2. If \(\infty\not\in\{W(0)\}\), then there exists a positive number \(r_0\) such that, for \(\lambda>1\) and \(r>r_0\), \[ T(r,W')<A\frac{\lambda v}{\lambda-1}T(\lambda r,W)\quad\text{ and }\quad T (r,W)<B\frac {\lambda v}{\lambda-1}\log\frac{e\lambda}{\lambda-1}T (\lambda r,W') \] with positive absolute constants \(A\) and \(B\). As a corollary he gets that \(W\) and \(W'\) have the same lower order.
growth of algebroid functions and their derivatives
Nevanlinna theory for algebroid functions
scientific article; zbMATH DE number 5563859