An algebroid function and its derivative (Q1023013)

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.