Common Borel radius of an algebroid function and its derivative

Abstract

In this article, by comparing the characteristic functions, we prove that for any $\nu$-valued algebroid function $w(z)$ defined in the unit disk with $\limsup_{r\to1-}T(r,w)/\log\frac{1}{1-r}=\infty$ and the hyper order $\rho_2(w)=0$, the distribution of the Borel radius of $w(z)$ and $w'(z)$ is the same. This is the extension of G. Valiron's conjecture for the meromorphic functions defined in $\widehat{\mathbb{C}}$.