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 ${\mathrm{limsup}}_{r\to 1-}T(r,w)/\log \frac{1}{1-r}=\infty$ and the hyper order ${\rho }_2(w)=0$, the distribution of the Borel radius of $w(z)$ and $w^{\prime }(z)$ is the same. This is the extension of G. Valiron's conjecture for the meromorphic functions defined in $\hat{\mathbb{C}}$.